4.Show that the following vector fields F are conservatives and findALL f such that F = Vf.=(a)(b)(2xy+z³)i+x²j+3xz²ki+y)² XF(x, y, z) =F(x, y) ={(x, y) = R² | x > y}.(xYX: - y)2jfor all (x, y, z) € R³.for all (x, y) E U :=

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Answered: Show that the following vector fields F…

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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Q.3) Graph the vector field F(x,y) 3j + zk and G(x, Y, z) = xi – yj+ zk. = -yi. Find V.(F × G) where F(x, y, z) = xi – -
6. Denote by V the vector ·(). Then we can define the operations grad (f) = f (gradient of f), div (F) = V F (divergence of F), and curl (F) = ▼ × F (curl of F). (a) Given F = (2x + 3y, 3x + z, sin x), calculate div(F) and curl(F). (b) Is the vector field F = (x + y,3y, z-x+1) conservative? What about F = (yz, xz, xy)? (c) Calculate V × (Vf) =curl(grad f) and V. (V × F) = div(curl F).
Match each of the following three vector fields to one of the four vector fields graphed below (yes, one graph does not have a match), and then explain your thinking: 1. (a) F(x, y) = (2y, 2.r). Match (circle one): I II III IV (b) F(x, y) = (x², 2y). Match (circle one): I II III IV (c) F(x, y) = (x², y²). Match (circle one): I II III IV (d) Explain your choices. Explanation:
The vector field F(x,y)= can be written as: O1. F(x,y)=(xy+x2 )i - (8)j O II. F(x,y)=(xy+x² )i + (2-10)j O II. (x,y)=(xy-2)i + (x² -10)j O IV. F(x,y)=x(y+x)i + (2 -10)j
Consider the vector field F(x,y) 2iu²j and the real-valued function f(x, y) = tan¬1(÷). One has V f(r,y) af it j dy (A) .... dx (-) (B) 1+(÷)² 1+ Since (-) i+ (C) Y 1+() 1+ly x2 + y? x2 + it follows that Vƒ = F. Hence F is conservative. Identify where the error(s) is (are) in the foregoing argument and give a brief explanation in words while perhaps also citing any relevant equation label(s) if appropriate in laying out your explanation.
Which of the following vector fields are conservative? (i) F(x, y) = (10x²y⁹+6)i + (9x³y² + 8) j (ii) F(x, y) = (9ye³x + cos 6y)i + (e⁹x + 6x sin 6y) j (iii) F(x, y) = 10y²e10xy i+ (10+xy) e10xy j

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Show that the following vector fields F are conservatives and find ALL f such that F = Vf. (a) (b) F(x, y, z) = (2xy+z³)i+x²j+3xz²k F(x, y) {(x, y) = R² | x > y}. == Y (x - y)² şi + X (x − y)2j for all (x, y, z) = R³. for all (x,y) ≤ U :=