7. Show that if the vector field F = Pi + Qj + Rk is conservative and P,Q, R havecontinuous first-order partial derivatives, then%3DdeӘРƏRaQƏRdyazdzдуHint: Recall that Clairaut's theorem states that if h(x, y, z) is a function whose secondpartials are all continuous, then hry = hyæ, hazhzz, and hyz = hzy.

Answered: 7. Show that if the vector field F = Pi…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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13- a) b) O лу z=f(x,y) = cos(2-2x + 2) + x²-x² COS Which one of the following is the gradient vector of f(x, y) at the point P = (0, 1)? (2-1) i + 1/2 j (-7/2+1) - 7/1 (2-1)1-5/1 (-7/2-1) 1 + 7/1 • (-/-1) i-7/1
26
1. (Section 17.1) Match the vector fields with their sketches below by placing the letter of the function in the corresponding blank: (I) F(x, y) = (II) F(x, y) = (III) F(x, y) = (IV) F(x, y) = Vector Field: (B) Vector Field: (A) (C) Vector Field (D) 0 Vector Field:
5) P = (−2, − 1) and draw the gradient vector at P. Draw to scale. Sketch the level curve of f(x, y) = x² - y² that passes through :
3. Let V denote the vector space of all functions f : R" → R, equipped with addition + : V × V → V defined via (ƒ+g)(x) = f(x)+ g(x), x ≤ R", and scalar multiplication : R × V → V defined via (\ · ƒ)(x) = \ƒ(x), ● x ER". Rº Now let W = {ƒ : R^ → R : ƒ(x) = ax + b for some a, b € R}, i.e. the space of all linear functions R" → R. (a) Find a basis for W. You should prove that it is indeed a basis.
Show that there is no vector field G = (P,Q, R) where P, Q and R have continuous second order partial derivatives such that curl G = (r", y, 2:"). %3D
1. Consider two vector fields: F1(x, y) = -yi+ xj and F2(7) = F. (a) Evaluate F and F, at the given points. (x, y) (1,0) (0, 1) (-1,0) (0, –1) F(r, y) F(1, y) (т, у) (1,1) (-1,1) (-1,–1) (1, –1) F(1, y) F(x, y) On the grids shown below, sketch above vectors of vector fields F (b) and F. F;(x, y) = -yi + xj F2(F) = F.
3. Show that the vector field F = -C with C> 0, is conservative and find its potential.
3. (b) Let f(x, y) = log((ry - 2)²), u =(³, 4). i. Calculate the directional derivative Vaf (3, 1). ii. Determine the unit vector (direction) u for which Vaf (3, 1) is maximal.
Which of the following pairs of functions is linearly dependent?
What are V-F and VX F for the vector field F(x,y,z) = 3x²zi-ye*j - y²zk? OV.F = x²exy 2 VXF = x²i-3e²j+yk OF=3x²-xe* + 2yz VXF = 2i-3j-k V-F = 3x²-x-z VXF= -2yi+ 3xj + zk V-F = 6xz-e* - y² VXF = -2yzi + 3x²j-ye*k

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