Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver

Size: px
Start display at page:

Download "Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver"

Transcription

1 Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin Narvik 6

2 PART I Task. Consider two-point boundary value problem D): { u x) = x for < x < u) = ; u) =. a) Show that the solution u is also the solution of a variational problem V). Derive the integral identity. Multiplying the equation by the test-function v V, integrating over [, ], we obtain u x)vx) dx = x vx) dx, and finally integrating by parts and using the fact that v) = v) =, we deduce u x)v x) dx + u )v) u )v) = which is valid for any function v. The formulation is: Find u V such that u, v ) = x, v) v V, u x)v x) dx = x vx) dx, where f, g) = fx)gx) dx. b) Show that the solution u is also a solution of the minimization problem M). Find the functional and formulate the minimization problem.

3 The formulation is: Find u V such that Fu) Fv) for any v V, where Fv) = v, v ) x, v). The solution u of the variational problem V ) is also a solution of the problem M) since the variational formulation and the minimization formulation are equivalent see the problem below). c) Prove the equivalence of these formulations, i.e. M) V ) D). By the tasks a) and b) we proved that D) = V ), D) = M) Now let us check that V ) = M). Assume that u is a solution of V), let v V and set w = v u so that v = u + w and w V. We have Fv) = Fu + w) = u + w, u + w ) x, u + w) = = u, u ) x, u) + u, w ) x, w) + w, w ) = = Fu) + + w, w ) Fu), since u, w ) x, w) = and w, w ). Let us show that M) = V ). Assume that u is a solution of M), let v V and denote by gt) the function gt) = Fu + tv) = u, u ) + tu, v ) + t v, v ) x, u) tx, v). 3

4 The differentiable function gt) has a minimum at t = and hence g ) =. It is easy to see that and hence u is a solution of V). Summing up, we have shown that g ) = u, v ) x, v) D) = V ) M). Finally, if u is a smooth weak solution, then from the integral identity of V) integrating by parts in the back direction we can obtain the equation of D). Everything is proved. d) Show that a solution to V) is uniquely determined. Suppose that we have two solutions u and u, i.e. u, v ) = x, v) u, v ) = x, v), v V. Subtracting these equations and choosing v = u u V, we get From this formula we have or u u ) dx =. u u ) = u x) u x) = C = const x [, ]. But from the boundary conditions u ) = u ) = and hence C =. Thus we checked that u x) = u x) x [, ]. e) Show that u u h ) ) dx 4 u v) ) dx

5 for any v V h. Here V h = {v V : v is piecewise linear functions} and u h is the approximate solution of the respective variational problem V h ). Recall that u is a solution of D) respectively V) ) and u h is a solution of V h ) that is u h, v ) = x, v) v V h. Subtracting the integral identities for V) and V h ) we obtain u u h ), v ) = v V h. ) Let v V h be an arbitrary function and set w = u h v. Then w V h and using ) with v replaced by w, we get, using Cauchy-Schwarz-Bunyakovski s inequality Dividing by u u h ) ) dx = u u h ), u u h ) ) + u u h ), w ) = = u u h ), u u h + w) ) = u u h ), u v) ) u u h ) ) dx u v) ) dx u u h ) ) dx we obtain the statement. Task. Consider boundary value problem D): { u x) = x for < x < u) = ; u ) =. a) Show that the solution u is also the solution of a variational problem V). Derive the integral identity.,. 5

6 Multiplying the equation by the test-function v V, integrating over [, ], we obtain u x)vx) dx = x vx) dx, and finally integrating by parts and using the fact that v) = u ) =, we deduce u x)v x) dx + u )v) u )v) = u x)v x) dx = x vx) dx, which is valid for any function v. The formulation is: Find u V such that u, v ) = x, v) v V, where f, g) = fx)gx) dx. The solution u of the variational problem V ) is also a solution of the problem M) since the variational formulation and the minimization formulation are equivalent see the problem below). b) Show that the solution u is also the solution of a minimization problem M). Find the functional and formulate the minimization problem. The formulation is: Find u V such that Fu) Fv) for any v V, where Fv) = v, v ) x, v). c) Show that u u h ) ) dx u v) ) dx 6

7 for any v V h. Here V h = {v V : v is piecewise linear functions} and u h is the approximate solution of the respective variational problem V h ). Let v V h be an arbitrary function and set w = u h v. Then w V h and using ) with v replaced by w, we get, using Cauchy-Schwarz- Bunyakovski s inequality Dividing by u u h ) ) dx = u u h ), u u h ) ) + u u h ), w ) = = u u h ), u u h + w) ) = u u h ), u v) ) u u h ) ) dx u v) ) dx u u h ) ) dx we obtain the statement. Task 3. Definition. A sequence {u k }, u k V is called the minimization sequence for the functional F if for any ε > there exists a number K such that Fu k ) < Fu) + ε k > K, where u gives a minimum to the functional. Using the fact that and keeping in mind that ) ) a b a + b + = a + b Fu) = u, u ) f, u),,. 7

8 prove that for a minimization sequence we have an estimate u l u m < ε l, m > K. Let us calculate. Note that f, u l ) f, u m ) + f, u ) l + u m =. u l u m u = l u m, u l ) u m = 4 u l, u l) u l, u m) + 4 u m, u m) = = u l, u l ) + u m, u m ) 4 u l, u l ) 4 u m, u m ) u l, u m ) = = u l, u l) + u u m, u m) l + u m, u l + ) u m f, u l ) f, u m ) + f, u ) ) l + u m ul + u m = Fu l ) + Fu m ) F < < Fu) + ε + Fu) + ε Fu) = ε. Here we used the following: Fu l ) < Fu) + ε l > K; Fu m ) < Fu) + ε m > K; ) ul + u m F Fu) l, m because u gives a minimum. Task 4. Consider the boundary value problem { d 4 u dx 4 = B, for < x <, u) = u ) = u) = u ) =. 8

9 Determine the approximate solution in the case of two intervals, assuming that the space W h consists on piecewise cubic functions. Compare with the exact solution. Let us remind that there exists only two basis functions for the partition onto two subintervals in the space W h see figures) Figure : First basis function. Figure : Second basis function. It is possible to calculate them. They are cubic then they have the form on each subinterval. The first function ϕ x) = ax 3 + bx + cx + d { ax 3 + bx + cx + d on, ) αx 3 + βx + γx + δ on, ) 9

10 satisfies the conditions: ϕ ) =, ϕ ) =, ϕ ) =, ϕ ) =, ϕ ) =, ϕ ) =. After the calculations we find { 6x ϕ x) = 3 + x on, ) 6x 3 36x + 4x 4 on, ) The second function { ax ϕ x) = 3 + bx + cx + d on, ) αx 3 + βx + γx + δ on, ) satisfies the conditions: ϕ ) =, ϕ ) =, ϕ ) =, ϕ ) =, ϕ ) =, ϕ ) =. After the calculations we find { 4x ϕ x) = 3 x on, ) 4x 3 x + 8x on, ) Now we calculate the stiffness matrix ϕ A =, ϕ ) ϕ, ϕ ϕ, ϕ ) ϕ, ϕ ) ) ). ϕ, ϕ ) = 4 ϕ, ϕ ) = 6 ϕ, ϕ ) = 96 4x) dx + 4 6x ) dx + 6 4x 3) dx = 4 3 = 9, 4x)6x )dx x 5) dx = 6, 4x 3)6x 5)dx =,

11 and the load vector b = B, ϕ ) B, ϕ ) ) B, ϕ ) = B B, ϕ ) = B 6x 3 + x )dx + B 4x 3 x )dx + B The corresponding linear system of equations has the form 9 6 A ξ = b ) ξ ξ 6x 3 36x + 4x 4)dx = B, 4x 3 x + 8x )dx =. ) = B and ξ = B 384, ξ =. Substituting these constants obtained coefficients) into the linear combination u h = ξ ϕ + ξ ϕ, we get, 4x 3 +, 3x on, u h B ),, 4x 3, 9x +, 6x, on, ), ) The exact solution u = Bx + c, u = B x + cx + c, u = B x3 6 + cx + c x + c,

12 u = B x4 4 + cx3 6 + c x + c x + c 3. From the boundary conditions we deduce c 3 = c =, c = B, c = B and finally ux) = 4 Bx4 Bx3 + 4 Bx. Task 5. Show that v P 3, ) is uniquely determined by the values v), v ), v), v ). To determine a cubic polynom, which has the form a 3 x 3 + a x + a x + a, in the unique way it is necessary to have four different conditions. Let us write the given conditions and check that they are different. We have a a + a + a = v) 3a 3 + a + a = v ) a a + a + a = v) 3a 3 + a + a = v ) They are different if det A, where A is a matrix of the system. The determinant of the matrix A is equal the the following: =. 3 Then there exists only one solution of the system and consequently the cubic polynom is determined uniquely.

13 Task 6. Consider the problem D) { u x) = x for < x < u) = ; u) =. And assume that V h is a finite-dimensional space of piecewise linear functions. Moreover the partition of the interval consists of M points. a) Formulate the problem V h ), which corresponds to problem D) in terms of stiffness matrix, load vector and coefficients of unknown function. The variational formulation in V h is V h ) Find u h V h : u h, v ) = x, v) v V h. It is easy to prove that instead of considering this identity for any v V h one can consider only M equations with basic functions u h, ϕ ) = x, ϕ ) u h, ϕ M ) = x, ϕ M ) Let us prove that from the system of equations the integral identity follows for any v V h. Suppose that the representation of v has the form: then v = η ϕ η M ϕ M, u h, v ) = u h, η ϕ η Mϕ M ) = η u h, ϕ ) η Mu h, ϕ M ) = = η x, ϕ ) η M x, ϕ M ) = x, η ϕ η M ϕ M ) = x, v). And we complete the proof. Now let us consider the representation of an unknown function u h = ξ ϕ ξ M ϕ M and substitute it in the system ). We get ξ ϕ ξ Mϕ M, ϕ ) = x, ϕ ) ξ ϕ ξ M ϕ M, ϕ M) = x, ϕ M ) 3 )

14 or ξ ϕ, ϕ ) ξ M ϕ M, ϕ ) = x, ϕ ) ξ ϕ, ϕ M ) ξ Mϕ M, ϕ M ) = x, ϕ M ). Finally the variational problem was reformulated in the form. Find unknown vector ξ, which satisfy the problem 3) Aξ = b, Here the stiffness matrix A is the matrix of system 3) and the load vector b is the vector of the right-hand-sides of system 3). b) Formulate the problem M h ), which corresponds to problem D) in terms of stiffness matrix, load vector and coefficients of unknown function. From the representation we have v = η ϕ η M ϕ M, av, v) = aη ϕ η M ϕ M, η ϕ η M ϕ M ) = = η aϕ, ϕ )η + η aϕ, ϕ )η η M aϕ M, ϕ M )η M = η Aη, Lv) = x, η ϕ η M ϕ M ) = b η, where the dot denotes the usual scalar inner) product in IR M : ζ η = ζ η ζ M η M. Minimization problem may be formulated as: Find unknown vector ξ IR M, such that [ ] ξ Aξ b ξ = min η IR M η Aη b η. 4

15 Task 7. Consider piecewise linear finite element space V h with basis elements {ϕ j x)}. Find the element stiffness matrix for the triangle K with vertices at, ),, ),, ). Without loss of generality let us denote by ϕ the function which is equal to in the point, ), by ϕ the function which is equal to in the point, ) and by ϕ 3 the function which is equal to in the point, ). The element stiffness matrix has the form where a K ϕ, ϕ ) a K ϕ, ϕ ) a K ϕ, ϕ 3 ) a K ϕ, ϕ ) a K ϕ, ϕ ) a K ϕ, ϕ 3 ) a K ϕ 3, ϕ ) a K ϕ 3, ϕ ) a K ϕ 3, ϕ 3 ) a K ϕ i, ϕ j ) = ϕ i ϕ j dx. K, Let us calculate the gradient of each basic functions. In fact they are ϕ = ), ϕ = Finally the element stiffness matrix is equal to ), ϕ 3 =. ). 5

16 PART II Task 8. Consider the convection diffusion problem { 5 u + u x + u x + u = x + x in ; u = on Γ. a) Derive the variational formulation V), which corresponds to this problem. By multiplying the equation by a test function v V = H ), integrating over and using the Green s formula for the Laplace term as usual, we get the following: v 5 u + u + u ) + u dx = x + x )v dx, x x then 5 u v + u + u ) ) + u v dx = x x x + x )v dx. Respectively, the variational formulation has the form Find u V such that au, v) = Lv) v V, where au, v) = 5 u v + Lv) = x + x )v dx. u + u ) ) + u v dx, x x b) Derive the minimization formulation M), which corresponds to this problem. 6

17 There is no associated minimization problems because the bilinear form is not symmetric. c) Verify V -ellipticity and continuity of the bilinear form and continuity of the linear form. Let us check V -ellipticity. By the Green s formula we have v v + v ) v dx = v n + n ) ds x x v v + v v ) x x From which we get the following: v + v ) vdx. 4) x x dx. Using 4) we rewrite the bilinear form as follows: av, v) = 5 v v + = v + v ) ) + v v dx = x x 5 v v + v ) dx v H ), i.e. α =. Now the continuity of the linear form. By the Cauchy-Schwarz-Bunyakovski s inequality we obtain Lv) = x + x )v dx x + x ) dx v dx and Λ = x + x ) L ). x + x ) L ) v H ) 7

18 Finally, the continuity of the bilinear form. By the Cauchy-Schwarz-Bunyakovski s inequality we get au, v) = 5 u v + u + u ) ) + u v x x dx ) u 5 u L ) v L ) + dx v x dx+ ) u + dx v x dx + u dx v dx 5 u H ) v H ) + u H ) v H )+ i.e. γ = 8. + u H ) v H ) + u H ) v H ) = 8 u H ) v H ), Task 9. Consider the convection diffusion problem { u + u x u x + u = sinx + x ) in ; u = on Γ. a) Derive the variational formulation V), which corresponds to this problem. By multiplying the equation by a test function v V = H ), integrating over and using the Green s formula for the Laplace term as usual, we get the following: v u + u u ) + u dx = sinx x x + x ) v dx, 8

19 then u v + u u ) ) + u v dx = x x sinx + x ) v dx. Respectively, the variational formulation has the form Find u V such that au, v) = Lv) v V, where au, v) = u v + u u ) ) + u v dx, x x Lv) = sinx + x ) v dx. b) Verify V -ellipticity and continuity of the bilinear form and continuity of the linear form. Let us check V -ellipticity. By the Green s formula we have v v v ) v dx = x x From which we obtain 4) and hence, i.e. α =. av, v) = v v + = v v v v ) x x v n + n ) ds dx. v v ) ) + v v dx = x x v v + v ) dx = v H ), 9

20 Now the continuity of the linear form. By the Cauchy-Schwarz-Bunyakovski s inequality we obtain Lv) = sinx + x ) v dx sin x + x ) dx v dx sinx + x ) L ) v H ) and Λ = sinx + x ) L ). Finally, the continuity of the bilinear form. By the Cauchy-Schwarz-Bunyakovski s inequality we get au, v) = u v + u u ) ) + u v x x dx ) u u L ) v L ) + dx v x dx+ ) u + dx v x dx + u dx v dx u H ) v H ) + u H ) v H )+ i.e. γ = 4. + u H ) v H ) + u H ) v H ) = 4 u H ) v H ), Task. Consider some rectangular finite elements. Let K be a rectangle with sides parallel to the coordinate axis in IR, for simplicity we consider K = [, ] [, ]. a) Find the number of element degrees of freedom for biquadratic functions Q K)).

21 A general form of the biquadratic function of two variables is vx) = a ij x i xj = i,j= = a +a x +a x +a x x +a x +a x +a x x +a x x +a x x. We have 9 unknown coefficients. Hence 9 element degrees of freedom. b) Prove that a function v Q K) is uniquely determined by the values at the vertices, midpoints of the sides and the value at the midpoint of the rectangle. Assume that we have two different functions v and v ) with the same given values in the nod-points, mid-points and in the center of the square. Consider the difference v = v v. It vanishes in all the 9 points of the square. Consider one side of the square I = {x IR : x =, x }. Quadratic function on this side is equal to a + a x + a x an it vanishes in three different points. Hence, it is identically equal to zero, i.e. a + a x + a x and we can factor out the function x, i.e. vx) = x w x, x ), where w is quadratic with respect to x and linear with respect to x. It is easy to calculate that w = a + a x + a x + a x x + a x + a x x. Consider the second side of the square I = {x IR : x =, x }. Quadratic function on the side vanishes in three points. Hence

22 it is identically equal to zero on this side. Hence we can factor out the function x, i.e. vx) = x x )w x, x ), where w is a bilinear function. Consider the third side of the square I = {x IR : x =, x }. Quadratic function on the side vanishes in three points. Hence it is identically equal to zero on this side. Hence we can factor out the function x, i.e. vx) = x x ) x )w 3 x, x ), where w 3 is a linear function with respect to x. Consider the last side of the square I = {x IR : x =, x }. Quadratic function on the side vanishes in three points. Hence it is identically equal to zero on this side. Hence we can factor out the function x, i.e. vx) = x x ) x )x w 4, where w 4 is a constant. Then, let us consider the value of v in the center of the square. From one hand v, =, ) from the other hand v, ) Hence w 4 = and v, i.e. v v. = w 4. c) Find one basis element if the vertices have the following coordinates:, ),, ),, ),, ), for instance, ψ x) = {, x =, x =, otherwise

23 This function is equal identically to zero on the sides I and I, hence we can factor out the function x x ), i.e. ψ x) = x x )a + a x + a x + a x x ). In the point, ) we have in the point, ) we have a + a =, a + a ) =, in the point, ) we have and in the point, ) we have a + a + a + a ) = 4 a + a + a + 4 a ) =. Solving this system of algebraic equations, we deduce a =, a =, a =, a = 4 or vx) = x x ) + x + x 4x x ). Task. Let πv P I) be the linear interpolant that agrees with v C I) at the end points of the segment, where I = [, h]. Prove that v πv L I) h max v L I). Denote the basis functions on the segment by ψ x) := h x h 3

24 and ψ x) := x h. A general function z P I) then has the following representation: hence, in particular zx) = z)ψ x) + zh)ψ x), x I, πvx) = v)ψ x) + vh)ψ x), x I, 5) since πv) = v), πvh) = vh). Using the Taylor expansion at x I: vy) = vx) + v x)y x) + Rx, y), where Rx, y) = v ξ)y x) is the remainder term of order and ξ I is a fixed point. In particular by taking y = and y = h, we get where v) = vx) + p) + Rx, ), vh) = vx) + ph) + Rx, h), 6) p) = v x)x, ph) = v x)h x). It is easy to see that x h and h x h, if x I. Hence, the estimate for the remainder term is Combining 5) and 6) we obtain Rx, ) h max v L I), Rx, h) h max v L I). πvx) = vx) ψ x) + ψ x)) + +p)ψ x) + ph)ψ x) + Rx, )ψ x) + Rx, h)ψ x), x I. 7) In the analysis we need the following Lemma: 4

25 Lemma For x I the following identities ψ x) + ψ x), 8) are valid. p)ψ x) + ph)ψ x) 9) By 7) and Lemma we get πvx) = vx) + Rx, )ψ x) + Rx, h)ψ x), x I and hence, vx) πvx) = Rx, )ψ x) Rx, h)ψ x). Keeping in mind 8) and the estimates for the remainder term, we deduce vx) πvx) Rx, ) ψ x) + Rx, h) ψ x) max { Rx, ), Rx, h )} ψ x) + ψ x)) h max v L I), x I, which leads to the estimate v πv L I) h max v L I). Proof of Lemma. To prove both statements we can use direct calculations: and ψ x) + ψ x) = x h + h x h h h = p)ψ x) + ph)ψ x) = v x)x ψ x) + v x)h x) ψ x) = = v x)x h x h + v x)h x) x h. 5

Høgskolen i Narvik Sivilingeniørutdanningen

Høgskolen i Narvik Sivilingeniørutdanningen Høgskolen i Narvik Sivilingeniørutdanningen Eksamen i Faget STE67 ELEMENTMETODEN Klasse: 4.ID Dato: 4.2.23 Tid: Kl. 9. 2. Tillatte hjelpemidler under eksamen: Kalkulator. Bok Numerical solution of partial

More information

Høgskolen i Narvik Sivilingeniørutdanningen

Høgskolen i Narvik Sivilingeniørutdanningen Høgskolen i Narvik Sivilingeniørutdanningen Eksamen i Faget STE6237 ELEMENTMETODEN Klassen: 4.ID 4.IT Dato: 8.8.25 Tid: Kl. 9. 2. Tillatte hjelpemidler under eksamen: Kalkulator. Bok Numerical solution

More information

Høgskolen i Narvik Sivilingeniørutdanningen

Høgskolen i Narvik Sivilingeniørutdanningen Høgskolen i Narvik Sivilingeniørutdanningen Eksamen i Faget STE66 ELASTISITETSTEORI Klasse: 4.ID Dato: 7.0.009 Tid: Kl. 09.00 1.00 Tillatte hjelpemidler under eksamen: Kalkulator Kopi av Boken Mechanics

More information

Inner Product Spaces

Inner Product Spaces Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

More information

MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.

MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α

More information

Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

More information

1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

More information

3. INNER PRODUCT SPACES

3. INNER PRODUCT SPACES . INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

More information

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v, 1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

More information

The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation. Lectures INF2320 p. 1/88 The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

THREE DIMENSIONAL GEOMETRY

THREE DIMENSIONAL GEOMETRY Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

More information

Numerical Solutions to Differential Equations

Numerical Solutions to Differential Equations Numerical Solutions to Differential Equations Lecture Notes The Finite Element Method #2 Peter Blomgren, blomgren.peter@gmail.com Department of Mathematics and Statistics Dynamical Systems Group Computational

More information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Recall that two vectors in are perpendicular or orthogonal provided that their dot Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

More information

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if 1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c

More information

Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points

Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a

More information

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

More information

since by using a computer we are limited to the use of elementary arithmetic operations

since by using a computer we are limited to the use of elementary arithmetic operations > 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations

More information

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

More information

Tangent and normal lines to conics

Tangent and normal lines to conics 4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

More information

DETERMINANTS. b 2. x 2

DETERMINANTS. b 2. x 2 DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

More information

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

More information

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)

More information

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

More information

FINITE DIFFERENCE METHODS

FINITE DIFFERENCE METHODS FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to

More information

College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions

College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use

More information

TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

More information

Vector and Matrix Norms

Vector and Matrix Norms Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

More information

Introduction to the Finite Element Method (FEM)

Introduction to the Finite Element Method (FEM) Introduction to the Finite Element Method (FEM) ecture First and Second Order One Dimensional Shape Functions Dr. J. Dean Discretisation Consider the temperature distribution along the one-dimensional

More information

1 Inner Products and Norms on Real Vector Spaces

1 Inner Products and Norms on Real Vector Spaces Math 373: Principles Techniques of Applied Mathematics Spring 29 The 2 Inner Product 1 Inner Products Norms on Real Vector Spaces Recall that an inner product on a real vector space V is a function from

More information

Summary of week 8 (Lectures 22, 23 and 24)

Summary of week 8 (Lectures 22, 23 and 24) WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

More information

SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

More information

NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

More information

FINITE ELEMENT : MATRIX FORMULATION. Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 7633

FINITE ELEMENT : MATRIX FORMULATION. Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 7633 FINITE ELEMENT : MATRIX FORMULATION Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 76 FINITE ELEMENT : MATRIX FORMULATION Discrete vs continuous Element type Polynomial approximation

More information

7 - Linear Transformations

7 - Linear Transformations 7 - Linear Transformations Mathematics has as its objects of study sets with various structures. These sets include sets of numbers (such as the integers, rationals, reals, and complexes) whose structure

More information

Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors 1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

More information

UNIVERSITETET I OSLO

UNIVERSITETET I OSLO NIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet Examination in: Trial exam Partial differential equations and Sobolev spaces I. Day of examination: November 18. 2009. Examination hours:

More information

minimal polyonomial Example

minimal polyonomial Example Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

More information

Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 126 Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

More information

A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors

More information

MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 425, PRACTICE FINAL EXAM SOLUTIONS. MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

More information

A Note on Di erential Calculus in R n by James Hebda August 2010.

A Note on Di erential Calculus in R n by James Hebda August 2010. A Note on Di erential Calculus in n by James Hebda August 2010 I. Partial Derivatives o Functions Let : U! be a real valued unction deined in an open neighborhood U o the point a =(a 1,...,a n ) in the

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

1 Completeness of a Set of Eigenfunctions. Lecturer: Naoki Saito Scribe: Alexander Sheynis/Allen Xue. May 3, 2007. 1.1 The Neumann Boundary Condition

1 Completeness of a Set of Eigenfunctions. Lecturer: Naoki Saito Scribe: Alexander Sheynis/Allen Xue. May 3, 2007. 1.1 The Neumann Boundary Condition MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations Lecture 11: Laplacian Eigenvalue Problems for General Domains III. Completeness of a Set of Eigenfunctions and the Justification

More information

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

More information

3.5 Spline interpolation

3.5 Spline interpolation 3.5 Spline interpolation Given a tabulated function f k = f(x k ), k = 0,... N, a spline is a polynomial between each pair of tabulated points, but one whose coefficients are determined slightly non-locally.

More information

Advanced Algebra 2. I. Equations and Inequalities

Advanced Algebra 2. I. Equations and Inequalities Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers

More information

Inner product. Definition of inner product

Inner product. Definition of inner product Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product

More information

Section 1.1. Introduction to R n

Section 1.1. Introduction to R n The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

More information

Galerkin Approximations and Finite Element Methods

Galerkin Approximations and Finite Element Methods Galerkin Approximations and Finite Element Methods Ricardo G. Durán 1 1 Departamento de Matemática, Facultad de Ciencias Exactas, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina. Chapter 1 Galerkin

More information

Equations, Inequalities & Partial Fractions

Equations, Inequalities & Partial Fractions Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

More information

4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

More information

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective

More information

A characterization of trace zero symmetric nonnegative 5x5 matrices

A characterization of trace zero symmetric nonnegative 5x5 matrices A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the

More information

Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series

Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series 1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4) ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

More information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

More information

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

More information

Time: 1 hour 10 minutes

Time: 1 hour 10 minutes [C00/SQP48] Higher Time: hour 0 minutes Mathematics Units, and 3 Paper (Non-calculator) Specimen Question Paper (Revised) for use in and after 004 NATIONAL QUALIFICATIONS Read Carefully Calculators may

More information

LECTURE NOTES: FINITE ELEMENT METHOD

LECTURE NOTES: FINITE ELEMENT METHOD LECTURE NOTES: FINITE ELEMENT METHOD AXEL MÅLQVIST. Motivation The finite element method has two main strengths... Geometry. Very complex geometries can be used. This is probably the main reason why finite

More information

Section 4.4 Inner Product Spaces

Section 4.4 Inner Product Spaces Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer

More information

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

More information

LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

More information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information

Section 1.2. Angles and the Dot Product. The Calculus of Functions of Several Variables

Section 1.2. Angles and the Dot Product. The Calculus of Functions of Several Variables The Calculus of Functions of Several Variables Section 1.2 Angles and the Dot Product Suppose x = (x 1, x 2 ) and y = (y 1, y 2 ) are two vectors in R 2, neither of which is the zero vector 0. Let α and

More information

α = u v. In other words, Orthogonal Projection

α = u v. In other words, Orthogonal Projection Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

More information

Linear Least Squares

Linear Least Squares Linear Least Squares Suppose we are given a set of data points {(x i,f i )}, i = 1,...,n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. One

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 01

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 01 Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 01 Welcome to the series of lectures, on finite element analysis. Before I start,

More information

Inner products on R n, and more

Inner products on R n, and more Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +

More information

Methods of Solution of Selected Differential Equations Carol A. Edwards Chandler-Gilbert Community College

Methods of Solution of Selected Differential Equations Carol A. Edwards Chandler-Gilbert Community College Methods of Solution of Selected Differential Equations Carol A. Edwards Chandler-Gilbert Community College Equations of Order One: Mdx + Ndy = 0 1. Separate variables. 2. M, N homogeneous of same degree:

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

The Not-Formula Book for C1

The Not-Formula Book for C1 Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

More information

4.3 Lagrange Approximation

4.3 Lagrange Approximation 206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

More information

1 Orthogonal projections and the approximation

1 Orthogonal projections and the approximation Math 1512 Fall 2010 Notes on least squares approximation Given n data points (x 1, y 1 ),..., (x n, y n ), we would like to find the line L, with an equation of the form y = mx + b, which is the best fit

More information

MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

More information

Linear Algebra: Vectors

Linear Algebra: Vectors A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector

More information

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

BALTIC OLYMPIAD IN INFORMATICS Stockholm, April 18-22, 2009 Page 1 of?? ENG rectangle. Rectangle

BALTIC OLYMPIAD IN INFORMATICS Stockholm, April 18-22, 2009 Page 1 of?? ENG rectangle. Rectangle Page 1 of?? ENG rectangle Rectangle Spoiler Solution of SQUARE For start, let s solve a similar looking easier task: find the area of the largest square. All we have to do is pick two points A and B and

More information

Mean Value Coordinates

Mean Value Coordinates Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its

More information

Chapter 7. Induction and Recursion. Part 1. Mathematical Induction

Chapter 7. Induction and Recursion. Part 1. Mathematical Induction Chapter 7. Induction and Recursion Part 1. Mathematical Induction The principle of mathematical induction is this: to establish an infinite sequence of propositions P 1, P 2, P 3,..., P n,... (or, simply

More information

T ( a i x i ) = a i T (x i ).

T ( a i x i ) = a i T (x i ). Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

More information

5 Indefinite integral

5 Indefinite integral 5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

More information

A gentle introduction to the Finite Element Method. Francisco Javier Sayas

A gentle introduction to the Finite Element Method. Francisco Javier Sayas A gentle introduction to the Finite Element Method Francisco Javier Sayas 2008 An introduction If you haven t been hiding under a stone during your studies of engineering, mathematics or physics, it is

More information

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

More information

Factoring Cubic Polynomials

Factoring Cubic Polynomials Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than

More information

Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:

Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1

More information

Lecture 14: Section 3.3

Lecture 14: Section 3.3 Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in

More information

Tim Kerins. Leaving Certificate Honours Maths - Algebra. Tim Kerins. the date

Tim Kerins. Leaving Certificate Honours Maths - Algebra. Tim Kerins. the date Leaving Certificate Honours Maths - Algebra the date Chapter 1 Algebra This is an important portion of the course. As well as generally accounting for 2 3 questions in examination it is the basis for many

More information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information

Equations Involving Lines and Planes Standard equations for lines in space

Equations Involving Lines and Planes Standard equations for lines in space Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity

More information