Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are
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Chapter 16 Solutions
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- Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].arrow_forwardFind an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}arrow_forwardFind the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).arrow_forward
- Find a basis for R2 that includes the vector (2,2).arrow_forwardDetermine whether each of the following vector fields F is path independent (conservative) or not. If it is path independent, enter a potential function for it. If it is path. dependent, enter NONE. (a) If F(x, y) = (5x² - y², -2xy), then f(x, y) = (b) If F(x, y) f(x, y) = = 2y x + 5 -i+2ln(x + 5), then (c) If F(x, y) = 3y sin(xy)i + 3x sin(xy)j, then f(x, y) =arrow_forward1. Let V = C[-2, 2], the vector space of continuous functions on the closed interval [-2, 2]. Let f, g E V and define the inner product of f and g to be (f, g) = | f(x)g(x) dx. Let h(x) = x? and let k(x) = 7x*, compute (h, k). Simplify your answer completely.arrow_forward
- 3. Let V denote the vector space of all functions ƒ : R" → R, equipped with addition + : V × V → V defined f: via (f+g)(x) = f(x) + g(x), x = R", and scalar multiplication : Rx V → V defined via (A. f)(x) = \f(x), ● x ER". Now let W = {f: R" → R: f(x) = ax + b for some a, b ≤ R}, i.e. the space of all linear functions R" → R. (a) Show that W is a subspace of V. (You may assume that V is a vector space). (b) Find a basis for W. You should prove that it is indeed a basis.arrow_forward8arrow_forwardLet f be a function of two variables that has continuous partial derivatives and consider the points A(5, 2), B(13, 2), C(5, 13), and D(14, 14). The directional derivative of f at A in the direction of the vector AB is 4 and the directional derivative at A in the direction of AC is 9. Find the directional derivative of f at A in the direction of the vector AD.arrow_forward
- , Let C" (R) be the vector space of "smooth" functions, i.e., real-valued functions f(x) in the variable z that have infinitely many derivatives at all points x E R. Let D: C* (IR) → C¤(R) and D² : C∞ (R) → C°(R) be the linear transformations defined by the first derivative D(f(x)) = f'(x) and the second derivative D²(f(x)) = f"(x). a. Determine whether the smooth function g(x) = 7e1z is an eigenvector of D. If so, give the associated eigenvalue. If not, enter NONE. Eigenvalue = b. Determine whether the smooth function h(x) = sin(9x) is an eigenvector of D2. If so, give the associated eigenvalue. If not, enter %3D NONE. Eigenvalue =arrow_forwardDetermine whether each of the following vector fields F is path independent (conservative) or not. If it is path independent, enter a potential function for it. If it is path dependent, enter NONE. – y?, –2xy), then (a) If F(x, y) = (4x² f(x, y) = 4y (b) If F(2x, y) i + 4 ln(x + 5)j, then f(x, y) : %| (c) If F(x, y) = 2y sin(xy)i + 2x sin(xy)j, then f(x, y) =arrow_forward1. Sketch the given vector fields by drawing some representative vectors. (a) F(x, y) = (x − y) î+xî (b) F(x, y) = (y, - x) /x² + y² (c) F(x, y, z) = 2y ĵarrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning