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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 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 x 2 x n It is often convenient to define formally the differential operator in vector form as: ( =,,...,. (7.2 x 1 x 2 x n Then we may view the gradient of ϕ, as the notation ϕ suggests, as the result of multiplying the vector by the scalar field ϕ. Note that the order ϕ of multiplication matters, i.e., x j is not ϕ x j. Let us now review a couple of facts about the gradient. For any j n, ϕ x j is identically zero on D iff ϕ(x 1, x 2,..., x n is independent of x j. Consequently, Moreover, for any scalar c, we have: ϕ = 0 on D ϕ = constant. (7.3 ϕ is normal to the level set L c (ϕ. (7.4 Thus ϕ gives the direction of steepest change of ϕ. 1

2 7.2 Divergence Let f : D R n, D R n, be a differentiable vector field. (Note that both spaces are n-dimensional. Let f 1, f 2,..., f n be the component (scalar fields of f. The divergence of f is defined to be div(f = f = n j=1 f j x j. (7.5 This can be reexpressed symbolically in terms of the dot product as f = (,..., (f 1,..., f n. (7.6 x 1 x n Note that div(f is a scalar field. Given any n n matrix A = (a ij, its trace is defined to be: tr(a = n a ii. i=1 Then it is easy to see that, if Df denotes the Jacobian matrix, then f = tr(df. (7.7 Let ϕ be a twice differentiable scalar field. Then its Laplacian is defined to be It follows from (7.1,(7.5,(7.6 that 2 ϕ = 2 ϕ x ϕ = ( ϕ. ( ϕ x ϕ. (7.9 x 2 n One says that ϕ is harmonic iff 2 ϕ = 0. Note that we can formally consider the dot product Then we have = ( x 1,..., (,..., x n x 1 x n = n 2 x 2 j=1 j. (7.10 2

3 2 ϕ = ( ϕ. (7.11 Examples of harmonic functions: (i D = R 2 ; ϕ(x, y = e x cos y. Then ϕ = x ex cos y, ϕ = y ex sin y, and 2 ϕ = e x cos y, 2 ϕ = e x cos y. So, 2 ϕ = 0. x 2 y 2 (ii D = R 2 {0}; ϕ(x, y = log( x 2 + y 2 = log(r. Then ϕ = x, ϕ = y, 2 ϕ = (x2 +y 2 2x(2x = (x2 y 2, and 2 ϕ = x x 2 +y 2 y x 2 +y 2 x 2 (x 2 +y 2 2 (x 2 +y 2 2 y 2 (x 2 +y 2 2y(2y = (x2 y 2. So, 2 ϕ = 0. (x 2 +y 2 2 (x 2 +y 2 2 These last two examples are special cases of the fact, which we mention without proof, that for any function f : D C which is differentiable in the complex sense, the real and imaginary part, R(f and I(f, are harmonic functions. Here f is differentiable in the complex sense if its total derivative Df at a point z D, a priori a R-linear map from C to itself, is in fact given by multiplication with a complex number, which we then call f (z. More concretely, ( this means that the matrix of Df in the basis 1, i is of the form a b b a for some real numbers a, b. We then have f (z = a + bi. There is a large supply of such functions since any f given (locally by a convergent power series in z is complex differentiable. In (i we can take f(z = e z = e x+iy = e x cos(y + ie x sin(y and in (ii we can take f(z = log(z = log(re iθ = log(r + iθ but we must be careful about the domain. To have a well defined argument θ for all z D we must make a cut in the plane and can only define f on, for example, D = {z = x + iy y = 0 x > 0} or D = {z = x + iy y = 0 x < 0}. But the union of D and D is C {0} as in (ii. (iii D = R n {0}; ϕ(x 1, x 2,..., x n = (x x x 2 n α/2 = r α for some fixed α R. Then ϕ x i = αr α 1 x i = r αrα 2 x i, and = α(α 2r α 4 x i x i + αr α ϕ x 2 i Hence 2 φ = n i=1 (α(α 2rα 4 x 2 i + αr α 2 = α(α 2 + nr α 2. So φ is harmonic for α = 0 or α = 2 n (α = 1 for n = 3. 3

4 7.3 Cross product in R 3 The three-dimensional space is very special in that it admits a vector product, often called the cross product. Let i,j,k denote the standard basis of R 3. Then, for all pairs of vectors v = xi + yj + zk and v = x i + y j + z k, the cross product is defined by v v = det x y z x y z = (yz y zi (xz x zj + (xy x yk. (7.12 Lemma 1 (a v v = v v (anti-commutativity (b i j = k, j k = i, k i = j (c v (v v = v (v v = 0. Corollary: v v = 0. Proof of Lemma (a v v is obtained by interchanging the second and third rows of the matrix whose determinant gives v v. Thus v v= v v. (b i j = det, which is k as asserted. The other two identities are similar. (c v (v v = x(yz y z y(xz x z + z(xy x y = 0. Similarly for v (v v. Geometrically, v v can, thanks to the Lemma, be interpreted as follows. Consider the plane P in R 3 defined by v,v. Then v v will lie along the normal line to this plane at the origin, and its orientation is given by the right hand rule: If the fingers of your right hand grab a pole and you view them from the top as a circle in the v v -plane that is oriented counterclockwise (i.e. corresponding to the ordering (v, v of the basis then the thumb points in the direction of v v. Finally the length v v is equal to the area of the parallelogram spanned by v and v. Indeed this area is equal to the volume of the parallelepiped spanned by v, v and a unit vector u = (u x, u y, u z orthogonal to v and v. We can take u = v v / v v and the (signed volume equals u x u y u z det x y z =u x (yz y z u y (xz x z + u z (xy x y x y z = v v (u 2 x + u 2 y + u 2 z = v v. 4

5 More generally, the same argument shows that the (signed volume of the parallelepiped spanned by any three vectors u, v, v is u (v v. 7.4 Curl of vector fields in R 3 Let f : D R 3, D R 3 be a differentiable vector field. Denote by P,Q,R its coordinate scalar fields, so that f = P i + Qj + Rk. Then the curl of f is defined to be: curl(f = f = det x y z P Q R. (7.13 Note that it makes sense to denote it f, as it is formally the cross product of with f. If the vector field f represents the flow of a fluid, then the curl measures how the flow rotates the vectors, whence its name. Proposition 1 Let h (resp. f be a C 2 scalar (resp. vector field. Then (a ( h = 0. (b ( f = 0. Proof: (a By definition of gradient and curl, = ( h = det x y z h x h y h z ( ( ( 2 h yz 2 h 2 h i + zy zx 2 h 2 h j + xz xy 2 h k. yx Since h is C 2, its second mixed partial derivatives are independent of the order in which the partial derivatives are computed. Thus, ( f = 0. (b By the definition of divergence and curl, ( f = ( x, y, z ( R y Q z, R x + P z, Q x P y 5

6 ( ( ( 2 R = xy 2 Q + 2 R xz yx + 2 P 2 Q + yz zx 2 P. zy Again, since f is C 2, 2 R Done. = 2 R xy yx, etc., and we get the assertion. Warning: There exist twice differentiable scalar (resp. vector fields h (resp. f, which are not C 2, for which (a (resp. (b does not hold. When the vector field f represents fluid flow, it is often called irrotational when its curl is 0. If this flow describes the movement of water in a stream, for example, to be irrotational means that a small boat being pulled by the flow will not rotate about its axis. We will see later in this chapter the condition f = 0 occurs naturally in a purely mathematical setting as well. Examples: (i Let D = R 3 {0} and f(x, y, z = i x (x 2 +y 2 that f is irrotational. Indeed, by the definition of curl, = z f = det x y z 0 y (x 2 +y 2 x (x 2 +y 2 y (x 2 +y 2 j. Show ( x i + ( ( ( y x j + ( y k x 2 + y 2 z x 2 + y 2 x x 2 + y 2 y x 2 + y 2 = [ ] (x 2 + y 2 + 2x 2 (x2 + y 2 2y 2 k = 0. (x 2 + y 2 2 (x 2 + y 2 2 (ii Let m be any integer 3, D = R 3 {0}, and f(x, y, z = 1 (xi + yj + zk, where r = x r 2 + y 2 + z 2. Show that f is not m the curl of another vector field. Indeed, suppose f = g. Then, since f is C 1, g will be C 2, and by the Proposition proved above, f = ( g would be zero. But, ( f = x, y, ( x z r, y m r, z m r m = rm 2x 2 ( m 2 rm 2 r 2m + rm 2y 2 ( m 2 rm 2 r 2m 6 + rm 2z 2 ( m 2 rm 2 r 2m

7 = 1 ( 3r m m(x 2 + y 2 + z 2 r m 2 = 1 (3 m. r 2m rm This is non-zero as m 3. So f is not a curl. Warning: It may be true that the divergence of f is zero, but f is still not a curl. In fact this happens in example (ii above if we allow m = 3. We cannot treat this case, however, without establishing Stoke s theorem. 7.5 An interpretation of Green s theorem via the curl Recall that Green s theorem for a plane region Φ with boundary a piecewise C 1 Jordan curve C says that, given any C 1 vector field g = (P, Q on an open set D containing Φ, we have: ( Q x P dx dy = P dx + Q dy. (7.14 y Φ We will now interpret the term Q P x y C. To do that, we think of the plane as sitting in R 3 as {z = 0}, and define a C 1 vector field f on D := {(x, y, z R( 3 (x, y D} by setting f(x, y, z = g(x, y = P i + Qj. Then i j k ( Q f = det x y z = P k, because P = Q = 0. Thus we y x z z P Q 0 get: ( f k = Q y P x. (7.15 And Green s theorem becomes: Theorem 1 ( f k dx dy = P dx + Q dy Φ C 7.6 A criterion for being conservative via the curl Here we just reformulate the remark after Ch. 6, Cor. 1 (which we didn t completely prove but just made plausible using the curl. 7

8 Proposition 1 Let g : D R 2, D R 2 open and simply connected, g = (P, Q, be a C 1 vector field. Set f(x, y, z = g(x, y, for all (x, y, z R 3 with (x, y D. Suppose f = 0. Then g is conservative on D. Proof: Since f = 0, Theorem 1 implies that P dx + Q dy = 0 for all C Jordan curves C contained in D. In fact, f = 0 also implies that P dx + Q dy = 0 for all closed curves but we won t prove this. Hence f is C conservative. Done. Example: D = R 2 {(x, 0 R 2 x 0}, g(x, y = Determine if g is conservative on D: y x 2 +y 2 i x j. x 2 +y 2 Again, define f(x, y, z to be g(x, y for all (x, y, z in R 3 such that (x, y D. Since g is evidently C 1, f will be C 1 as well. By the Proposition above, it will suffice to check if f is irrotational, i.e., f = 0, on D R. This was already shown in Example (i of section 4 of this chapter. So g is conservative. 8

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

16.5: CURL AND DIVERGENCE

16.5: CURL AND DIVERGENCE 16.5: URL AN IVERGENE KIAM HEONG KWA 1. url Let F = P i + Qj + Rk be a vector field on a solid region R 3. If all first-order partial derivatives of P, Q, and R exist, then the curl of F on is the vector

More information

The Vector or Cross Product

The Vector or Cross Product The Vector or ross Product 1 ppendix The Vector or ross Product We saw in ppendix that the dot product of two vectors is a scalar quantity that is a maximum when the two vectors are parallel and is zero

More information

Math 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z).

Math 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z). Math 29 Solutions to Assignment 7. Find the gradient vector field of the following functions: a fx, y lnx + 2y; b fx, y, z x cosy/z. Solution. a f x x + 2y, f 2 y x + 2y. Thus, the gradient vector field

More information

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors. 3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with

More information

v 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)

v 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product) 0.1 Cross Product The dot product of two vectors is a scalar, a number in R. Next we will define the cross product of two vectors in 3-space. This time the outcome will be a vector in 3-space. Definition

More information

Differentiation of vectors

Differentiation of vectors Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where

More information

AB2.2: Curves. Gradient of a Scalar Field

AB2.2: Curves. Gradient of a Scalar Field AB2.2: Curves. Gradient of a Scalar Field Parametric representation of a curve A a curve C in space can be represented by a vector function r(t) = [x(t), y(t), z(t)] = x(t)i + y(t)j + z(t)k This is called

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

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:

More information

52. The Del Operator: Divergence and Curl

52. The Del Operator: Divergence and Curl 52. The Del Operator: Divergence and Curl Let F(x, y, z) = M(x, y, z), N(x, y, z), P(x, y, z) be a vector field in R 3. The del operator is represented by the symbol, and is written = x, y, z, or = x,

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

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

MAT 1341: REVIEW II SANGHOON BAEK

MAT 1341: REVIEW II SANGHOON BAEK MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and

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

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

More information

3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field

3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important

More information

Scalar Valued Functions of Several Variables; the Gradient Vector

Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,

More information

5.3 The Cross Product in R 3

5.3 The Cross Product in R 3 53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

More information

Rotation Matrices and Homogeneous Transformations

Rotation Matrices and Homogeneous Transformations Rotation Matrices and Homogeneous Transformations A coordinate frame in an n-dimensional space is defined by n mutually orthogonal unit vectors. In particular, for a two-dimensional (2D) space, i.e., n

More information

1 Conservative vector fields

1 Conservative vector fields Math 241 - Calculus III Spring 2012, section CL1 16.5. Conservative vector fields in R 3 In these notes, we discuss conservative vector fields in 3 dimensions, and highlight the similarities and differences

More information

Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

More information

( ) ( ) 1. Let F = ( 1yz)i + ( 3xz) j + ( 9xy)k. Compute the following: A. div F. F = 1yz x. B. curl F. i j k. = 6xi 8yj+ 2zk F = z 1yz 3xz 9xy

( ) ( ) 1. Let F = ( 1yz)i + ( 3xz) j + ( 9xy)k. Compute the following: A. div F. F = 1yz x. B. curl F. i j k. = 6xi 8yj+ 2zk F = z 1yz 3xz 9xy . Let F = ( yz)i + ( 3xz) j + ( 9xy)k. Compute the following: A. div F F = yz x B. curl F + ( 3xz) y + ( 9xy) = 0 + 0 + 0 = 0 z F = i j k x y z yz 3xz 9xy = 6xi 8yj+ 2zk C. div curl F F = 6x x + ( 8y)

More information

3 Contour integrals and Cauchy s Theorem

3 Contour integrals and Cauchy s Theorem 3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of

More information

DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents

DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition

More information

2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

More information

Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product

Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.

More information

We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P. Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

More information

SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA

SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the

More information

dv div F = F da (1) D

dv div F = F da (1) D Note I.6 1 4 October The Gauss Theorem The Gauss, or divergence, theorem states that, if is a connected three-dimensional region in R 3 whose boundary is a closed, piece-wise connected surface and F is

More information

vector calculus 2 Learning outcomes

vector calculus 2 Learning outcomes 29 ontents vector calculus 2 1. Line integrals involving vectors 2. Surface and volume integrals 3. Integral vector theorems Learning outcomes In this Workbook you will learn how to integrate functions

More information

6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.

6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w. hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,

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

Dot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product

Dot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot

More information

Fundamental Theorems of Vector Calculus

Fundamental Theorems of Vector Calculus Fundamental Theorems of Vector Calculus We have studied the techniques for evaluating integrals over curves and surfaces. In the case of integrating over an interval on the real line, we were able to use

More information

Lecture L3 - Vectors, Matrices and Coordinate Transformations

Lecture L3 - Vectors, Matrices and Coordinate Transformations S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

More information

MthSc 206 Summer1 13 Goddard

MthSc 206 Summer1 13 Goddard 16.1 Vector Fields A vector field is a function that assigns to each point a vector. A vector field might represent the wind velocity at each point. For example, F(x, y) = y i+x j represents spinning around

More information

FINAL EXAM SOLUTIONS Math 21a, Spring 03

FINAL EXAM SOLUTIONS Math 21a, Spring 03 INAL EXAM SOLUIONS Math 21a, Spring 3 Name: Start by printing your name in the above box and check your section in the box to the left. MW1 Ken Chung MW1 Weiyang Qiu MW11 Oliver Knill h1 Mark Lucianovic

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

Solutions to Practice Final Exam

Solutions to Practice Final Exam Math V22. Calculus IV, ection, pring 27 olutions to Practice Final Exam Problem Consider the integral x 2 2x dy dx + 2x dy dx x 2 x (a) ketch the region of integration. olution: ee Figure. y (2, ) y =

More information

DERIVATIVES AS MATRICES; CHAIN RULE

DERIVATIVES AS MATRICES; CHAIN RULE DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we

More information

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain

More information

Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I. Ronald van Luijk, 2012 Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

More information

4B. Line Integrals in the Plane

4B. Line Integrals in the Plane 4. Line Integrals in the Plane 4A. Plane Vector Fields 4A-1 Describe geometrically how the vector fields determined by each of the following vector functions looks. Tell for each what the largest region

More information

VECTOR CALCULUS: USEFUL STUFF Revision of Basic Vectors

VECTOR CALCULUS: USEFUL STUFF Revision of Basic Vectors Prof. S.M. Tobias Jan 2009 VECTOR CALCULUS: USEFUL STUFF Revision of Basic Vectors A scalar is a physical quantity with magnitude only A vector is a physical quantity with magnitude and direction A unit

More information

Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation

Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7- In this section, we discuss linear transformations 89 9 CHAPTER

More information

F = 0. x ψ = y + z (1) y ψ = x + z (2) z ψ = x + y (3)

F = 0. x ψ = y + z (1) y ψ = x + z (2) z ψ = x + y (3) MATH 255 FINAL NAME: Instructions: You must include all the steps in your derivations/answers. Reduce answers as much as possible, but use exact arithmetic. Write neatly, please, and show all steps. Scientists

More information

Unified Lecture # 4 Vectors

Unified Lecture # 4 Vectors Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

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

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

LINEAR MAPS, THE TOTAL DERIVATIVE AND THE CHAIN RULE. Contents

LINEAR MAPS, THE TOTAL DERIVATIVE AND THE CHAIN RULE. Contents LINEAR MAPS, THE TOTAL DERIVATIVE AND THE CHAIN RULE ROBERT LIPSHITZ Abstract We will discuss the notion of linear maps and introduce the total derivative of a function f : R n R m as a linear map We will

More information

4.2. LINE INTEGRALS 1. 2 2 ; z = t. ; y = sin

4.2. LINE INTEGRALS 1. 2 2 ; z = t. ; y = sin 4.2. LINE INTEGRALS 1 4.2 Line Integrals MATH 294 FALL 1982 FINAL # 7 294FA82FQ7.tex 4.2.1 Consider the curve given parametrically by x = cos t t ; y = sin 2 2 ; z = t a) Determine the work done by the

More information

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z 28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition

More information

A matrix over a field F is a rectangular array of elements from F. The symbol

A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

More information

Again, the limit must be the same whichever direction we approach from; but now there is an infinity of possible directions.

Again, the limit must be the same whichever direction we approach from; but now there is an infinity of possible directions. Chapter 4 Complex Analysis 4.1 Complex Differentiation Recall the definition of differentiation for a real function f(x): f f(x + δx) f(x) (x) = lim. δx 0 δx In this definition, it is important that the

More information

Section 12.6: Directional Derivatives and the Gradient Vector

Section 12.6: Directional Derivatives and the Gradient Vector Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate

More information

x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0. Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability

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

13.4 THE CROSS PRODUCT

13.4 THE CROSS PRODUCT 710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product

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

88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a 88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

More information

Solutions for Review Problems

Solutions for Review Problems olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

More information

Chapter Eighteen. Stokes. Here also the so-called del operator = i + j + k x y z. i j k. x y z p q r

Chapter Eighteen. Stokes. Here also the so-called del operator = i + j + k x y z. i j k. x y z p q r hapter Eighteen tokes 181 tokes's Theorem Let F:D R be a nice vector function If the curl of F is defined by F( x, = p( x, i + q( x, j + r( x, k, r q p r q p curlf = i + j + k y z z x x y Here also the

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, 1982, 2008. This chapter originates from material used by author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals,

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

Determinants, Areas and Volumes

Determinants, Areas and Volumes Determinants, Areas and Volumes Theodore Voronov Part 2 Areas and Volumes The area of a two-dimensional object such as a region of the plane and the volume of a three-dimensional object such as a solid

More information

This makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5

This makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5 1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,

More information

NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

More information

If Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the Right-Hand-Rule.

If Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the Right-Hand-Rule. Oriented Surfaces and Flux Integrals Let be a surface that has a tangent plane at each of its nonboundary points. At such a point on the surface two unit normal vectors exist, and they have opposite directions.

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

Chapter 1 Vectors, lines, and planes

Chapter 1 Vectors, lines, and planes Simplify the following vector expressions: 1. a (a + b). (a + b) (a b) 3. (a b) (a + b) Chapter 1 Vectors, lines, planes 1. Recall that cross product distributes over addition, so a (a + b) = a a + a b.

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

Chapter 17. Review. 1. Vector Fields (Section 17.1)

Chapter 17. Review. 1. Vector Fields (Section 17.1) hapter 17 Review 1. Vector Fields (Section 17.1) There isn t much I can say in this section. Most of the material has to do with sketching vector fields. Please provide some explanation to support your

More information

15.1. Vector Analysis. Vector Fields. Objectives. Vector Fields. Vector Fields. Vector Fields. ! Understand the concept of a vector field.

15.1. Vector Analysis. Vector Fields. Objectives. Vector Fields. Vector Fields. Vector Fields. ! Understand the concept of a vector field. 15 Vector Analysis 15.1 Vector Fields Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives! Understand the concept of a vector field.! Determine

More information

Groups, Rings, and Fields. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, S S = {(x, y) x, y S}.

Groups, Rings, and Fields. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, S S = {(x, y) x, y S}. Groups, Rings, and Fields I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, A binary operation φ is a function, S S = {(x, y) x, y S}. φ : S S S. A binary

More information

Problem set on Cross Product

Problem set on Cross Product 1 Calculate the vector product of a and b given that a= 2i + j + k and b = i j k (Ans 3 j - 3 k ) 2 Calculate the vector product of i - j and i + j (Ans ) 3 Find the unit vectors that are perpendicular

More information

Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve. cosh y = x + sin y + cos y

Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve. cosh y = x + sin y + cos y Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve at the point (0, 0) is cosh y = x + sin y + cos y Answer : y = x Justification: The equation of the

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

Solutions to Practice Problems for Test 4

Solutions to Practice Problems for Test 4 olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,

More information

PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS

PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS PROBLEM SET Practice Problems for Exam #2 Math 2350, Fall 2004 Nov. 7, 2004 Corrected Nov. 10 ANSWERS i Problem 1. Consider the function f(x, y) = xy 2 sin(x 2 y). Find the partial derivatives f x, f y,

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

Chapter 2. Parameterized Curves in R 3

Chapter 2. Parameterized Curves in R 3 Chapter 2. Parameterized Curves in R 3 Def. A smooth curve in R 3 is a smooth map σ : (a, b) R 3. For each t (a, b), σ(t) R 3. As t increases from a to b, σ(t) traces out a curve in R 3. In terms of components,

More information

CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS

CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS CHAPTER 6: RATIONAL NUMBERS AND ORDERED FIELDS LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we construct the set of rational numbers Q using equivalence

More information

LINE INTEGRALS OF VECTOR FUNCTIONS: GREEN S THEOREM. Contents. 2. Green s Theorem 3

LINE INTEGRALS OF VECTOR FUNCTIONS: GREEN S THEOREM. Contents. 2. Green s Theorem 3 LINE INTEGRALS OF VETOR FUNTIONS: GREEN S THEOREM ontents 1. A differential criterion for conservative vector fields 1 2. Green s Theorem 3 1. A differential criterion for conservative vector fields We

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

Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere

Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x

More information

Practice Problems for Midterm 2

Practice Problems for Midterm 2 Practice Problems for Midterm () For each of the following, find and sketch the domain, find the range (unless otherwise indicated), and evaluate the function at the given point P : (a) f(x, y) = + 4 y,

More information

CBE 6333, R. Levicky 1. Potential Flow

CBE 6333, R. Levicky 1. Potential Flow CBE 6333, R. Levicky Part I. Theoretical Background. Potential Flow Potential Flow. Potential flow is irrotational flow. Irrotational flows are often characterized by negligible viscosity effects. Viscous

More information

( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those

( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those 1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make

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

VECTOR CALCULUS Stokes Theorem. In this section, we will learn about: The Stokes Theorem and using it to evaluate integrals.

VECTOR CALCULUS Stokes Theorem. In this section, we will learn about: The Stokes Theorem and using it to evaluate integrals. VECTOR CALCULU 16.8 tokes Theorem In this section, we will learn about: The tokes Theorem and using it to evaluate integrals. TOKE V. GREEN THEOREM tokes Theorem can be regarded as a higher-dimensional

More information

6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu) 6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,

More information

Review of Vector Analysis in Cartesian Coordinates

Review of Vector Analysis in Cartesian Coordinates R. evicky, CBE 6333 Review of Vector Analysis in Cartesian Coordinates Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers.

More information

AB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss

AB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss AB2.5: urfaces and urface Integrals. Divergence heorem of Gauss epresentations of surfaces or epresentation of a surface as projections on the xy- and xz-planes, etc. are For example, z = f(x, y), x =

More information

F Matrix Calculus F 1

F Matrix Calculus F 1 F Matrix Calculus F 1 Appendix F: MATRIX CALCULUS TABLE OF CONTENTS Page F1 Introduction F 3 F2 The Derivatives of Vector Functions F 3 F21 Derivative of Vector with Respect to Vector F 3 F22 Derivative

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

POWER SETS AND RELATIONS

POWER SETS AND RELATIONS POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

More information

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

More information