19. Consider the vector fieldV(r, y, 2) =1-y|x²+(y-1)² > x²+(y-1)²(y: H)and the curvesC1 : x² + (y – 1)2 = 1, 62 : x² + (y – 2)² = 4 and 63 : x² + (y – 4)² = 1.(a) ComputeV• dr.

Answered: Image /qna-images/answer/ed360ceb-f731-449a-9fea-25c7511096df.jpg

Answered: 19. Consider the vector field V(x, y,…

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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6. Vector fields V and W are defined by V %3D (2х — Зу + z, -3х — у + 4z, 4y + z) W%3D (2x — 4y — 5z, -4х + 2у, -5х + 62) Determine which vector field is conservative and denote it by F.
6. (a) Show that the curve represented by the vector function F(t) = (5t, 3t – 4, 2t² – 6) is at the intersection between a plane and a cylinder. (b) Find a vector function that represents the intersection of the sphere x² + y² + z² = 1 and the elliptic cylinder x2 + 2z2 = 1.
F(x,y,z) = ⟨1,y,z^4/3⟩ a. Find the parametric equation of a curve passing through the point (-1,1,1) which is parallel to the vector field F at all of its points, with the parameter t = x. b. Find the equation of a surface passing through the point (-1,1,1) which is perpendicular to the vector field F at all of its points.
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I'm not too sure how to calculate vector fields
1) F=Cy+2xz)i+ (x+ z²+1)j +(2yz+x)K Give the Vector field. Protected, ☆Fis the vector fied protected! Your show. If find a Potential function for the vector field "F".
5. Let 7 - 7" be the separation vector from a fixed point (x', y', z') to the point (x, y, z). Show that (a) V() = - デーデ (b) デーデ2 V(|7 – F"|?) = 2.
Find a scalar field, 0, whose gradient is the irrotational vector V = (x² - yz, y² — xz, ² - xy) In other words, find such that Vo=V₁ Note: • Remember that your tutorial has Hints in it, at the end of each chapter. If you're stuck, they are often quite useful! • If the answer is a scalar, you can just type it in the box (using the Calcpad if you like, or using / for fractions, ^ for exponents, shift and - for subscripts, etc.) For multiplication, you can either leave a space, or use *. So x* x = xx = x². Note that this is not the same as x without a space; that will get read as an entirely different variable! • If you need to enter a vector, enter an ordered list of components, so for A you can enter either A = (Az. Ay, A₂) = {Az, Ay, A₂}. Note that the system isn't great with multiplying through by overall factors, so it's better not to write e.g. o= (2A, 2A, 2A₂). + X .. 15 calc Opera Functi Symb Relati
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. (12) Let V = R². Define vector addition and scalar multiplication as follows: (u1, uz)O(v1, v2) = (u1V1, U2V2) and c(u1,u2) = (u,C, u2^), c > 0 (a) Does V have an additive identity? If so, what would it be? (b) Does the distributive axiom c(uOv) = cu@cv hold? (Show why or why not). 4
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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19. Consider the vector field V(x, y, z) = ( HÓ-DE) 1-y x²+(y–1)² » 2+(y-1)2 and the curves 8 : 2 + (y – 1)? =1, :2² + (y – 2)² = 4 and C3 : 2² + (y – 4)² = 1. C3 : x² + (y – 4)² = 1. | (a) Compute V• dr.