MATH 2400, Analytic Geometry and Calculus 3

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1 MATH 2400, Analytic Geometry and Calculus 3 List of important Definitions and Theorems 1 Foundations Definition 1. By a function f one understands a mathematical object consisting of (i) a set X, called the domain of the function, (ii) a set Y, called the range of the function, (iii) a set Γ of pairs (x, y) of points x X and y Y, called the graph of the function, such that for each x X there is a unique f(x) Y with ( x, f(x) ) Γ. To better denote the function f one writes f : X Y, x f(x). Example 2. The following are examples of functions: (1) the identity function on a set X, id X : X X, x x, (2) the square-root function : 0, x x, (3) the exponential function exp :, x exp(x), (4) the distance functions from the origin δ : 2, (x, y) x 2 + y 2 and δ : 3, (x, y, z) x 2 + y 2 + z 2. Definition 3. Functions of the form f :, x mx, f : 2, (x, y) mx + ny, f : 3, (x, y, z) mx + ny + lz where m, n, l, are called linear. Functions of the form f :, x mx + a, where a, m, n, l, are called affine. f : 2, (x, y) mx + ny + a, f : 3, (x, y, z) mx + ny + lz + a Definition 4. A formula of the form T 1 (x 1,..., x n ) = T 2 (x 1,..., x n ) is called an equation. Here T 1 (x 1,..., x n ), and T 2 (x 1,..., x n ) are terms built from the variables x 1,..., x n, numerical constants, and functions. A tuple of numbers (a 1,..., a n ) is said to satisfy this equation if the equation is true when we perform the substitution x 1 = a 1,..., x n = a n. The graph of an equation is the set of all tuples that satisfy it. 1

2 Example 5. (1) The equation x 2 + y 2 + z 2 = 9 equates two terms T 1 and T 2. Here T 1 is a term built from the variables x, y and z, the squaring function, and addition, while T 2 is the constant 9. The triple (0, 3, 0) satisfies this equation, while (25, 10, 3) does not. In general, a triple (a 1, b 1, c 1 ) satisfies this equation if it is a point on the surface of a sphere of radius 3 centered at the origin. (2) Let f be a function. The graph of the equation y = f(x) is the same as the graph of f. To see this, let ( x, f(x) ) Γ be an element of the graph of f. Substituting these values into the equation y = f(x) yields the equation f(x) = f(x), which is true. Furthermore, suppose (x, y) satisfies y = f(x). Then (x, y) is an element of the graph of f. Please note well that not every equation has a graph that is the graph of a function. 2 Limits and Continuity Definition 6. The function f : 2 has the limit L at the point (a, b) 2, written if for every ε > 0 there is a δ > 0 such that lim f(x, y) = L, (x,y) (a,b) f(x, y) L < ε for all (x, y) (a, b) with d ( (x, y), (a, b) ) < δ. emark 7. Intuitively, lim (x,y) (a,b) f(x, y) = L means that f(x, y) is as close to L as we wish whenever the distance of the point (x, y) to (a, b) is sufficiently small. Definition 8. The function f : 2 is called continuous at the point (a, b) 2, if lim f(x, y) = f(a, b). (x,y) (a,b) The function f is said to be continuous on a region 2, if it is continuous at every point (a, b). 2

3 3 Vectors Definition 9. The n-dimensional euclidian space n is defined as the set of all n-tupels (x 1,...,x n ), where x 1,...,x n. Elements of n are also called vectors. If x = (x 1,..., x n ) and y = (y 1,...,y n ) are two elements of n, the displacement vector with tail x and tip y is the vector (y 1 x 1,...,y n x n ) n. The following provides several operations on vectors in n-dimensional euclidian space. Definition 10. The sum of two vectors x = (x 1,...,x n ) and y = (y 1,...,y n ) is defined as x + y = (x 1 + y 1,..., x n + y n ). If λ is a scalar (i.e. an element of ), and x = (x 1,...,x n ) a vector, the scalar multiple of x by λ is defined by λ x = (λx 1,...,λx n ). If x = (x 1,...,x n ) and y = (y 1,...,y n ) are two vectors, their dot product or scalar product is defined as the real number x y = x 1 y x n y n. If x = (x 1, x 2, x 3 ) and y = (y 1, y 2, y 3 ) are two vectors of 3, their cross product is defined by x y = (x 2 y 3 x 3 y 2, x 3 y 1 x 1 y 3, x 1 y 2 x 2 y 1 ). 3

4 4 Differentiability Definition 11. A function f : 2, (x, y) f(x, y) is called partially differentiable in the point (a, b) 2 with respect to the variable x (resp. y), if the limit ( f f(a + h, b) f(a, b) (a, b) := lim x h 0 h resp. f (a, b) := lim h 0 f(a, b + h) f(a, b) h ) exists. One then calls f x f (a, b) and (a, b) the partial derivatives of f at (a, b). If f is partially differentiable in every point of 2 with respect to the variables x and y, then one says that f is partially differentiable. A function f : 2, (x, y) f(x, y) is called twice partially differentiable, if it is partially differentiable, and if the partial derivatives f f and are partially differentiable x as well. A function f : 2, (x, y) f(x, y) is called differentiable in the point (a, b) 2, if there exists a linear function L : 2 such that lim (x,y) (a,b) E(x, y) (x a)2 + (y b) 2 = 0, where E : 2 is the error function defined by E(x, y) := f(x, y) f(a, b) L(x, y). One then calls L the linear approximation of f at (a, b), and writes f(x, y) f(a, b) + L(x, y). Definition 12. A function f : n, x f(x) is called partially differentiable in the point a = (a 1,, a n ) n with respect to the variable x i, if for every i, 1 i n the limit f f(a 1,, a i + h,, a n ) f(a 1,, a n ) (a) := lim x i h 0 h exists. One then calls f x i (a) the (i-th) partial derivative of f at a with respect to the variable x i. If f is partially differentiable in every point of n with respect to the variables x 1,...,x n, then one says that f is partially differentiable. A function f : n, x f(x) is called twice partially differentiable, if it is partially differentiable, and if the partial derivatives f x i, 1 i n are partially differentiable as well. A function f : n, x f(x) is called differentiable in the point a n, if there exists a linear function L : n such that lim x a E(x) (x1 a 1 ) (x n a n ) 2 = 0, 4

5 where E : n is the error function defined by E(x) := f(x) f(a) L(x). One then calls L the linear approximation of f at a, and writes f(x) f(a) + L(x). Theorem 13. If f : n, x f(x) is differentiable at a n, then f is partially differentiable and continuous at a. Theorem 14. If f : n, x f(x) is twice partially differentiable, and the second partial derivatives 2 f := f x i x j x i x j are continuous, then for 1 i, j n. 2 f x i x j = 2 f x j x i Definition 15. If f : n, x f(x) is a partially differentiable function, and x n, the vector ( f gradf(x) := (x),, f ) (x) x 1 x n is called the gradient of f at x. Given a function f : n, x f(x) which is partially differentiable (up to isolated points), a point x n is called a critical point of f, if gradf(x) = 0, or if gradf(x) is not defined. 5

6 5 Local and Global Extrema Definition 16. If f : n, x f(x) is a function, a point a n is called a local maximum (resp. local minimum) of f, if f(x) f(a) (resp. f(x) f(a)) for all x n near a. The point a n is called a global maximum (resp. global minimum) of f over the region n, if f(x) f(a) (resp. f(x) f(a)) for all x. Theorem 17. Assume that f : 2, (x, y) f(x, y) is a twice continuously partially differentiable function. Suppose that (a, b) is a point where grad f(a, b) = 0. Let D = 2 f x (a, b) 2 f (a, b) 2 2 ( ) 2 2 f (a, b). x If D > 0 and 2 f (a, b) > 0, then f has a local minimum in a. x2 If D > 0 and 2 f (a, b) < 0, then f has a local maximum in a. x2 If D < 0, then f has a saddle point in a. If D = 0, no conclusion can be made: f can have a local maximum, a local minimum, a saddle point, or none of these in the point (a, b). Definition 18. If f : 2, (x, y) f(x, y) and g : 2, (x, y) g(x, y) are functions, and c a number, a point (a, b) 2 is called a local maximum (resp. local minimum) of f under the constraint g(x, y) = c, if f(x, y) f(a, b) (resp. f(x, y) f(a, b)) for all (x, y) 2 near (a, b) which satisfy g(x, y) = c. The point (a, b) 2 is called a global maximum (resp. global minimum) of f under the constraint g(x, y) = c, if f(x, y) f(a, b) (resp. f(x, y) f(a, b)) for all (x, y) 2 which satisfy g(x, y) = c. Theorem 19. Assume that f : 2, x f(x) is a smooth function, and g : 2, x g(x) a smooth constraint. If f has a maximum or minimum at the point (a, b) under the constraint g(x, y) = c, then (a, b) either satisfies the equations gradf(a, b) = λ gradg(a, b) and g(a, b) = c for some λ, or grad g(a, b) = 0. The number λ is called the Lagrange multiplier. 6

7 6 Coordinate Transformations The coordinate transformation from polar to cartesian coordinates on 2 is given by [0, ) [0, 2π] 2 ( ) ( ) x cosθ (, θ) = y sin θ. The coordinate transformation from cylindrical to cartesian coordinates on 3 is given by [0, ) [0, 2π] 3 x cosθ (, θ, z) y = sin θ z z The coordinate transformation from spherical to cartesian coordinates on 3 is given by. [0, ) [0, 2π] [0, π] 3 x sin φ cosθ (, θ, φ) y = sin φ sinθ z cos φ Definition 20. The Jacobian (x,y) of some coordinate transformation (s, t) ( x(s, t), y(s, t) ) (s,t) of 2 is defined as the determinant (x, y) (s, t) = x s s The Jacobian (x,y,z) of some coordinate transformation (s, t, u) ( x(s, t, u), y(s, t, u), z(s, t, u) ) (s,t,u) of 3 is defined as the determinant (x, y, z) (s, t, u) = x s s z s Theorem ( 21. ) If T 2 is a region which under the coordinate transformation (s, t) x(s, t), y(s, t) of 2 transforms to, and if f : is a continuous function, the following change of variables formula for integrals holds true: f(x, y) dxdy = f ( x(s, t), y(s, t) ) (x, y) (s, t) ds dt. T 7 x t t x t t z t. x u u z u..

8 Theorem ( 22. If T 3 is a ) region which under the coordinate transformation (s, t, u) x(s, t, u), y(s, t, u), z(s, t, u) of 3 transforms to, and if f : is a continuous function, the following change of variables formula for integrals holds true: f(x, y, z) dxdy dz = f ( x(s, t, u), y(s, t, u), z(s, t, u) ) (x, y, z) (s, t, u) ds dt du. T 8

9 7 Curves and Vector Fields Definition 23. By a curve in n one understands the image of a continuous map r : [a, b] n. The map r : [a, b] n is called a parametrization of the curve C. The curve C is called closed, if r(a) = r(b). Definition 24. A continuous map F : n, where n is a region, is a called a vector field on n. Definition 25. If F : n n is a vector field, a differentiable curve r : [a, b] n is called a flow line of F, if r (t) = F( r(t)) for all t [a, b]. Definition 26. A vector field F on an open region of n is called a gradient field, if F = grad(f) for a differentiable function f on. One then calls f a potential for F. Definition 27. The scalar curl of a vector field F = F 1 i + F 2 j : 2 over some region 2 is defined as curl F := F 2 x F 1. Definition 28. The curl of a vector field F = F 1 i+f 2 j +F 3 k : 3 over some region 3 is defined as ( curl F F3 := F ) ( 2 F1 i + z z F ) ( 3 F2 j + x x F ) 1 k. Theorem 29. If is an open convex (or contractible) region in 2 (resp. 3 ), and F a differentiable vector field on, then F is a gradient field if and only if curl F = 0. 9

10 8 Line integrals and Green s Theorem Definition 30. Let C be a piecewise smooth oriented curve in 2 (or 3 ), described by the (piecewise smooth) parametrization r : [a, b] C. Assume further that F is a continuous vector field on a region of 2 (resp. 3 ) containing C. Then the line integral of F along the curve C is given by C F d r = b a F( r(t)) r (t) dt. Theorem 31 (Fundamental Theorem of Calculus for Line Integrals). Let C be a piecewise smooth oriented curve in 2 (or 3 ) parametrized by r : [a, b] C, and let F = gradf be a gradient vector field defined on a region containing C. Then the line integral of F along the curve C depends only on f and the endpoints of C and is given by F d r = f( r(b)) f( r(a)). C Definition 32. A vector field F on an open region of n is called conservative, if for each closed curve C in the line integral of F along the curve C satisfies F d r = 0. C Theorem 33. A vector field F on an (open connected) region of n is a gradient vector field if and only if it is a conservative vector field. In that case, after fixing a point p, a potential is given by f(q) = F d r, C q where q, and C q is a piecewise smooth curve from p to q. Theorem 34 (Green s Theorem). Assume that C is a closed piecewise smooth curve in 2 which is the boundary of some region in 2 such that the region lies always to the left of the curve. Assume further that F is a smooth vector field on an open region containg C and. Then the following formula holds true: F d r = curl F da. C 10

11 9 Flux Integrals Definition 35. Let S be a smooth oriented surface in 3 having a smooth parametrization r : 3, (s, t) r(s, t), where is a region in 2. The flux of a smooth vector field F through S then is given by F da = F ( r(s, t) ) ( r s r ) ds dt. t S Theorem 36. Let S be a surface in 3 and F a smooth vector field defined on an open region containing S. (i) If S is the graph of a smooth function z = f(x, y) above a region in the xy-plane, the flux of F through S is F da = F ( x, y, f(x, y) ) ( f x i f ) j + k dxdy. S (ii) If S is a cylindrical surface around the z-axis of radius and oriented away from the z-axis, the flux of F through S is F da = F (, θ, z) ) ( ) cosθ i + sin θ j dz dθ, S where T is the θz-region corresponding to S. (iii) If S is a spherical surface of radius and oriented away from the origin, the flux of F through S is F da = F (, θ, φ) ) ( sin φ cosθ i + sin φ sin θ j + cosφ ) k 2 sin φ dθ dφ, S where T is the θφ-region corresponding to S. 10 Calculus of Vector Fields Definition 37. The divergence of a vector field F = F 1 i+f 2 j +F 3 k : 3 over some open region 3 is defined as div F := F 1 x + F 2 + F 3 z. Theorem 38 (The Divergence Theorem). Assume that W is a solid region in 3 whose boundary S is a piecewise smooth surface, and that F is a smooth vector field on an open region containing W and S. Then F da = div F dv, where S is given the outward orientation. S W 11

12 Theorem 39 (Stokes Theorem). Assume that S is a smooth oriented surface in 3 with piecewise smooth orientiable boundary C, and that F is a smooth vector field on an open region containing S and C. Then F d r = curl F da, where C is given the orientation induced by S. C Theorem 40. If is an open convex (or contractible) region in 3, and F a differentiable vector field on, then F is a curl field if and only if div F = 0. S 12