Suppose S and E satisfy the conditions of the Divergence Theorem and f is a scalar function with continuous partial derivatives. Prove that
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Multivariable Calculus
- 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_forwardWhich ONE of the following statements is TRUE? A. The directional derivative as a scalar quantity is always in the direction vector u with |u|=1. B. Gradient of f(x,y,z) at some point (a,b,c) is given by ai+bj+ck. OC. The directional derivative is a vector valued function in the direction of some point of the gradient of some given function. D. The cross product of the gradient and the uint vector of the directional vector gives us the directional derivative. E. None of the choices in this list.arrow_forward42. Derivatives of triple scalar products a. Show that if u, v, and w are differentiable vector functions of t, then du v X w + u• dt dv X w + u•v X dt dw (u•v X w) dt dt b. Show that d'r dr? dr dr d'r dt dt r. dt dr? (Hint: Differentiate on the left and look for vectors whose products are zero.)arrow_forward
- 1. Obtain the directional derivative of: a.. f(x,y) = x²-4x³y² at the point (1,-2) in the direction of a unit vector whose angle with the semi-axis x is 1, u = cos i + sen j b. f(x,y) = x²-xy + 3y² at the point (-1,-2) in the direction of a unit vector whose angle with the semi-axis x is u = cos 0 i+sen j c. f(x,y) = x²sin y at the point (1.1) in the direction of a vector v = 3i-4jarrow_forward1. Consider the function F(x, y, z) = (√/1 – x² − y², ln(e² — z²)). This function is a mapping from R" to Rm. Determine the values of m and n. (b) Is this function scalar-valued or vector-valued? Briefly explain. (c) Determine the domain and range of F and sketch the corresponding regions. (d) Is it possible to visualize this function as a graph? If so, sketch the graph of F.arrow_forward7. Properties of position vectors Let r = xi + yj + zk and let r = |r|. a. Show that Vr = r/r. b. Show that V(r") = nr"-2r. c. Find a function whose gradient equals r.arrow_forward
- 3-find the gradient of the function v if w=(xy)/z at the point(1,-2,1), find the directional derivative of w in the direction of vector v = 2i + j − 2k, then find the maximum value of the directional derivative.arrow_forwardConsider the following function. T: R2 → R, T(x, y) = (2x2, 3xy, y?) Find the following images for vectors u = (u,, u2) and v = (v,, v2) in R2 and the scalar c. (Give all answers in terms of 1' "1, U2, V1, V2, and c.) T(u) T(v) T(u) + T(v) = T(u + v) CT(u) = T(cu) = Determine whether the function is a linear transformation. O linear transformation not a linear transformationarrow_forwardB- Find the directional derivative of the function W = x² + xy + z³ at the point P: (2,1,1) in the direction towards P₂(5,4,2). əz Ju əv B- If Z = 4e* Iny, x = In(u cosv) and y = u sinv find andarrow_forward
- Calculus Questionarrow_forwardFind the rate of change of F at P in the direction of the vector ⟨2,-3,6⟩ and the minimum rate of change of F at P.arrow_forwardLet r1(t) and r2(t) be continuous vector functions with two components, and let t0 be a point in the domains of both r1 and r2. Prove thatlimt→t0(r1(t) · r2(t)) = r1(t0) · r 2(t0).arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning