A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

Transcription

1 A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry Definition of a Vector Scalar Product Properties of the Scalar Product Length and Unit Vectors Angle Parallel and Perpendicular Vectors Lines Planes ross Product Properties of the ross Product Distances of Points from a Plane in Other epresentations of Lines and planes in Spheres and ylinders 4 2. Parameterized urves in Velocity, Speed, and Acceleration Velocity of curves with r(t) = Smooth urves Arclength urvature Normal and Binormal Vectors Torsion Frenet Frame urvature of urves NOT parmeterized with respect to Arclength 5 3. Surfaces in n Domain and ange Limits ontinuity Partial Derivatives Differentiability Directional Derivaitve Gradient Vector Directional Derivatives and the Gradient Vector Higher Derivatives and lairaut s Theorem Global and Local Extrema ritical Points Functions of Two Variables Second Derivative Test The Double Integral Fubini s Theorem Level Surfaces Tangent Planes of Level Surfaces Using Lagrange Multipliers to Find Extrema with onstraints The Triple Integral hain ule 7 1

2 3.20. hange of Variables Formula in Two Dimensions Polar oordinates Surface Area Path Integrals 7 4. Vector Fields Line Integrals onservative Vector Fields and the Potential Function Line Integrals of onservative Vector Fields Line Integrals of Vector Fields in onservative Vector Fields in Green s Theorem 8 1. Analytic Geometry 1.1. Definition of a Vector. A vector v is an n-tuple of real numbers: v = (v 1,..., v n ). Given two vectors v, w n, addition and multiplication with a scalar t are defined by v + w = (v 1,..., v n ) + (w 1,..., w n ) = (v 1 + w 1,..., v n + w n ) t v = t (v 1,..., v n ) = (tv 1,..., tv n ). From the definitions, it follows immediately that addition and scalar multiplication of vectors are: (1) distributive: t (v + w) = t v + t w (2) associative: (u + v) + w = u + (v + w) (3) commutative: v + w = w + v 1.2. Scalar Product. The scalar product or dot product v w is defined by n v w = (v 1,..., v n ) (w 1,..., w n ) = v i w i Properties of the Scalar Product. It follows from the definition that the scalar product is (1) linear: t (v w) = (t v) w = v (t w), and u (v + w) = u v + u w (2) commutative: v w = w v 1.4. Length and Unit Vectors. The length of a vector v is defined by v = v v = v v2 n. Vectors of length 1 are called unit vectors Angle. The angle θ between two vector v and w is defined implicitly by v w = v w cos θ. For two vectors v, w with v, w 0, the enclosed angle θ is hence given by n θ = cos 1 i=1 v iw i ( n i=1 v2 i )1/2 ( n i=1 w2 i )1/2 This yields a geometric interpretation of the scalar product: to get v w, w is projected orthogonally onto v, and the length w cos θ of the projected vector is multiplies with v. 2 i=1

3 1.6. Parallel and Perpendicular Vectors. Two vectors v, w are called (1) perpendicular or orthogonal if v w = 0. We denote this by v w. (2) parallel if v w = v w. We denote this by v w. Note that two vectors v, w are parallel if and only if there is a scalar t such that v = t w. The zero vector 0 = (0,..., 0) is by definition parallel and perpendicular to every vector Lines. Let u, v be two vectors with v 0. Then yields a line in n. We say that v spans the line. u + t v = (u 1 + tv 1,..., u n + tv n ); t 1.8. Planes. let u, v, w n be vectors where v and w are not parallel. Then We say that v and w span the plane. u + s v + t w = (u 1 + sv 1 + tw 1,..., u n + sv n + tw n ) 1.9. ross Product. At a point p of any plane in 3 there is exactly one line perpendicular to the plane. If v and w span the plane, a vector spanning this perpendicular line is given by the cross product: For two vectors v, w 3 the cross product v w is defined by v w = (v 1, v 2, v 3 ) (w 1, w 2, w 3 ) = (v 2 w 3 v 3 w 2, v 3 w 1 v 1 w 3, v 1 w 2 v 2 w 1 ) Alternatively, the cross product is given by the 3 3 determinant v w = v i j k 1 v 2 v. 3 w 1 w 2 w 3 v w is perpendicular to both v and w v, w and v w are oriented according to the right hand rule if θ is the angle betweenv and w, then v w = v w sin θ. The length of the v w is equal to the area of the parallelogram spanned by v and w Properties of the ross Product. (1) linear: t (v w) = (t v) w = v (t w) and u (v + w) = u v + u w. (2) anti-commutative: v w = w v Another remarkable property of the cross product is the following: u (v w) = (u v) w. Geometrically, u (v w) is the volume of the parallelepiped given by u, v, and w. The notion of a perpendicular vector yields another form of representing a plane in 3 : Let r 0 = (x 0, y 0, z 0 ) be a fixed point in the plane and n a vector perpendicular to the plane. An arbitrary point r = (x, y, z) on the plane satisfies n (r r 0 ) = 0. Moreover, the equation n (r r 0 ) = 0 defines a plane in 3 for any fixed n, v 3, provided n Distances of Points from a Plane in 3. Let n, r 0 3 with n = 1 be given. Then n (r r 0 ) yields the distance from the point r = (x, y, z) to the plane which passes through r 0 and is perpendicular to n. Note that the plane s distance from the oriin in this case is given by n r 0. 3

4 1.12. Other epresentations of Lines and planes in 3. In 3, the representation of a line l(t) = (x, y, z) = u + t v can be solved for t in each coordinate if none of the values v 1, v 2, v 3 are zero: t = x u 1 v 1 t = y u 2 v 2 t = z u 3 v 3 This yields another representation of a line in 3, which is called a symmetric equation: x u 1 v 1 = y u 2 v 2 = z u 3 v 3. The parametric equation representing a plane can be transformed similarly giving the standard equation of a plane: ax + by + cz = d, where the normal vector n is (a, b, c) Spheres and ylinders. The vector notation allows for an elegant description of other geometric objects such as spheres and cylinders. For a scalar r 0 and a vector c 3, the equation x c 2 = r 2 yields a sphere of radius r centered at c. For a scalar r 0 and vectors c, n 3 with n = 1, the equation n (x c) 2 = r 2 yields a cylinder of radius r centered around the line given by c + t n. 2. Parameterized urves in Velocity, Speed, and Acceleration. Given three differentiable functions x, y, z : [a, b], the vector r(t) = (x(t), y(t), z(t)) can be understood as describing a particle moving through space; the particle s position depends on the (time) parameter t. This perception gives rise to the following definitions. We define the velocity of r at time t to be r (t) = (x (t), y (t), z (t)) speed of r at time t to be r(t) = [x (t)] 2 + [y (t)] 2 + [z (t)] 2 acceleration of r at time t to be r (t) = (x (t), y (t), z (t)) Velocity of curves with r(t) = 1. Suppose that r(t) = 1 for all t for curve r(t). Differentiating r(t) = 1, we observe that r (t) is orthogonal to r(t) for all t: r(t) = 1 r(t) r (t) = 0 for all t Smooth urves. A curve r : [a, b] 3 is called smooth if its speed vanishes at most at the endpoints: r (t) 0 for all t (a, b) Arclength. The arclength l(a, b) of a curve r : [a, b] 3 is given by l(a, b) = b a r (t) dt A curver : [a, b] is called parameterized with respect to arclength if r (t) = 1 for all t [a, b] urvature. Let r : [a, b] 3 be parameterized with respect to arclength. Then κ(t) = r (t) is called the curvature of r at t. Differentiating the equation r (t) = 1 shows that r (t) and r (t) are orthogonal for all t. If κ(t) 0, the vector is a well-defined unit vector. n(t) = r (t) r (t) 4

5 2.6. Normal and Binormal Vectors. If κ(t) 0, we can define the Normal vector of the curve at time t to be n(t) Binormal vector of the curve at time t to be b(t) = r (t) n(t) The binormal vector is obviously a unit vector, so we can apply the same reasoning as before to see that b(t) and b (t) are orthogonal. On the other hand, differentiating b(t) = r (t) n(t) we get: Hence b (t) is parallel to n(t). b (t) = r (t) n(t) + r (t) n (t) = r (t) n (t) Torsion. The equation b (t) = τ(t) n(t) defines the torsion τ of the curve at time t Frenet Frame. Whenever κ(t) 0, the vectors r (t), n(t), b(t) are mutually orthogonal unit vectors. They span the Frenet Frame urvature of urves NOT parmeterized with respect to Arclength. Let r : [a, b] 3 be amy smooth curve. Its curvature κ(t) at time t is given by κ(t) = r (t) r (t) r (t) 3 3. Surfaces in n 3.1. Domain and ange. A function f : n assigns a unique real value f(x 1,..., x n ) to each point (x 1,..., x n ) of a set D in n. The set D is called the domain of f, the set Imf = {f(x 1,..., x n ) (x 1,..., x n ) D} is called the image or range of f Limits. A real number L is said to be the limit of f at (a, b,... ) if for all sequences (a m, b m,... ) with lim m a m = a, lim m b m = b,..., the following holds: lim f(a m, b m,... ) = L. m We denote this by lim f(x 1, x 2,... ) = L. (x 1,x 2,... ) (a,b,... ) Equivalently, if for every real number ɛ > 0 there is another real number δ > 0 such that (x 1, x 2,... ) (a, b,... ) < δ f(x 1, x 2,... ) L < ɛ ontinuity. A function f : n with domain D is said to be continuous at (a, b,... ) D if lim f(x 1, x 2,... ) = f(a, b,... ) (x 1,x 2,... ) (a,b,... ) 3.4. Partial Derivatives. Let f : n be a function with domain D and (x 1,..., x n ) D. The partial derivative of f at (x 1,..., x n ) with respect to x i is given by the limit f f(x 1,..., x i + h,..., x n ) f(x 1,..., x i,..., x n ) = lim. x i h 0 h 3.5. Differentiability. A function f : n with domain D is called differentiable at (x 1,..., x n ) D if all partial derivatives exist and are continuous at (x 1,..., x n ) Directional Derivaitve. Let f : n be a function with domain D and r D. Suppose that u is a unit vector in n. The directional derivative of f at r in the direction of u is given by d f(r + tu) dt. t= Gradient Vector. Let f : n be a function with domain D that is differentiable at r D. The gradient vector f of f at r is given by f ( f r = x 1,..., f ) r x n r 5

6 3.8. Directional Derivatives and the Gradient Vector. Let f : n be a function with domain D that is differentiable at r D. Using the chain rule, the directional derivative is given by d f(r + tu) dt = f r u t= Higher Derivatives and lairaut s Theorem. Let f : n be a function with domain D and suppose the partial derivatives of f are themselves differentiable. Then differentiating f x i with respect to x j is the same as differentiating f x j with respect to x i : 2 f = 2 f. x i x j x j x i Global and Local Extrema. A function f : n with domain D has a global maximum at r D if f(x) f(r) for all x D a global minimum at r D if f(x) f(r) for all x D a local maximum at r D if there is a disc centered at r such that f(x) f(r) for all x a local minimum at r D if there is a disc centered at r such that f(x) f(r) for all x ritical Points. Let f : D n be differentiable. We call a point r D a critical point if f r = 0. If f has an extremum at r, then r is critical, but the converse is not necessarily true Functions of Two Variables Second Derivative Test. Let f : D 2 and its derivatives be differentiable and let (x 0, y 0 ) D be a critical point of f. Let ( 2 f 2 ( f 2 ) 2) D(x 0, y 0 ) = x 2 y 2 f x y (x0,y 0) if D(x 0, y 0 ) > 0 and 2 f x 2 < 0, then f has a maximum at (x 0, y 0 ). (x0,y 0) if D(x 0, y 0 ) > 0 and 2 f x 2 > 0, then f has a minimum at (x 0, y 0 ). (x0,y 0) if D(x 0, y 0 ) < 0, then f has a saddle at (x 0, y 0 ). if D(x 0, y 0 ) = 0, then the second derivative test gives no information about the nature of the critical point The Double Integral. Let = [a, b] [c, d] and let f : be continuous. The double integral of f over is defined to be f(x, y) da = lim (x i x i 1 )(y j y j 1 )f(x i, yj ) P 0 where P = P [a,b] P [c,d] is a partition of, P = P [a,b] P [c,d] is its norm, and (x i, y j ) [x i 1, x i ] [y j 1, y j ]. i,j Fubini s Theorem. Let f : 2 be continuous and = [a, b] [c, d]. The double integral is given by b d d b f(x, y) da = f(x, y) dydx = f(x, y) dxdy a c Level Surfaces. Let f(x, y, z) : 3 be a function of three variables. The level set {(x, y, z) f(x, y, z) = c} for a constant c generally yields a surface in 3. 6 c a

7 3.16. Tangent Planes of Level Surfaces. onsider the level set {(x, y, z) f(x, y, z) = c} of a function f : 3. (1) The gradient vector f(x 0, y 0, z 0 ) at a point (x 0, y 0, z 0 ) is perpendicular to the plane tangent to the level surface at (x 0, y 0, z 0 ). (2) The tangent plane is given by the equation f(x 0, y 0, z 0 ) (x x 0, y y 0, z z 0 ) = Using Lagrange Multipliers to Find Extrema with onstraints. let f : 3 be differentiable. To find the maximum and minimum value of f subject to the constraint g(x, y, z) = c, the gradients f and g must be parallel. An algorithm to find the maximum and minimum values is hence given by: (1) Find all points (x, y, z) such that f = λ g, for some λ, and g(x, y, z) = c. (2) Evaluate f at these points. The largest (smallest) value is the maximum (minimum) of f subject to the constraint g(x, y, z) = c The Triple Integral. Let g : 3 be continuous and = [a, b] [c, d] [e, f]. The triple integral of g over is defined to be g(x, y, z) dv = lim (x i x i 1 )(y j y j 1 )(z k z k 1 )g(x i, yj, zk) P 0 i,j,k hain ule. Let r(t) be a smooth curve in n and let f : n be a function of several variables. Then df dt = f r (t). More generally, suppose each of the variables x i is a function of the variables t 1,..., t m, then f t j = n i=1 f x i x i t j hange of Variables Formula in Two Dimensions. Let (x(u, v), y(u, v)) be a parameter transformation with Jacobian matrix ( ) J = x/ u x/ v y/ u y/ v which maps S 2 into 2. Then f(x, y) dxdy = S f(u, v) det J dudv Polar oordinates. The parameter transformation (x(r, θ), y(r, θ)) = (r cos θ, r sin θ) expresses (x, y) in polar coordinates. Then f(x, y) dxdy = f(r, θ) rdrdθ Surface Area. Let f : 2 be differentiable. The surface area of f over D is given by [ ] 2 [ ] 2 f f dxdy x y Path Integrals. Let f : 2 be differentiable and let r(t) parameterize a smooth curve in 2. The path integral of f along is given by f(r(t)) r (t) dt 7 S

8 4. Vector Fields A vector field F assigns to each point in a domain n a vector in n Line Integrals. Let F be a vector field on a domain n and let be a smooth curve in parameterized by r(t). The line integral of F along is given by F(r(t)) r (t) dt 4.2. onservative Vector Fields and the Potential Function. If F = f for some function f : 2, then F is called a conservative vector field with potential function f Line Integrals of onservative Vector Fields. Let F = f be conservative on a domain n. For any continuous curve in from u to b which is parameterized by r(t), we have F r (t) dt = f(v) f(u). In particular (1) If is closed then F r dt = 0. (2) if is another continuous curve in from u to v parameterized by s(t), then e F s dt = F r dt. The integral is said to be path-independent Line Integrals of Vector Fields in 2. Let F = (P, Q) be a vector field on a domain 2 and let be a smooth curve in given by (x(t), y(t)). Then b F(P (x(t), y(t)), Q(x(t), y(t))) (x (t), y (t)) dt = P (x, y) dx + Q(x, y) dy a 4.5. onservative Vector Fields in 2. If F = (P, Q) = f, then by lairaut s theorem, P y = Q x Green s Theorem. Let be a simply connected region with positively-oriented boundary. Then Q P dx + Q dy = x P y dxdy Dept. of Mathematics, Bard ollege, Annandale-On-Hudson, NY address: [email protected] 8