Suppose S is the union of the cylinder x² + y²at z = 1. Suppose is a vector field such that=1 for 0 ≤ z ≤ 1 and the disk x² + y² ≤ 1VXF= sinh(z)(x +y), zeu+cos(a+s), (2z+y)tanCalculate the flux of V x F though S.-¹(2)).

Given the surface S is the union of the cylinder x2+y2=1,0≤z≤1 and the disk x2+y2≤1 at z=1.Let us…

Answered: Suppose S is the union of the cylinder…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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5) Find the flux of F = (-2y, -x) on the circle of radius 2 centered at the origin. Use the vector field grapher shown in class and cut and paste the vector field here. Before you calculate the flux do you expect your answer to be positive, negative, or zero?
Find the divergence of the vector field V(x, y, z) = −3xi + (y + 4 cos(x))j + (9z + e5xy) k. (Give an exact answer. Use symbolic notation and fractions where needed.) div V =
9. Find the flux of the vector field B(x, y,z) shown in the left figure through the square S of side 2 shown in the right figure , oriented in the j direction, where B(x, v. z)== yi +xj x²+y² S + 2 1 3
For each of the following vector fields, find its curl and determine if it is a gradient field. (a) ♬ = 5yz i + (5xz + z²) j + (5xy+2yz) k curl F F? = (b) Ğ = (10xy + x³) i+ (5x² + z²) j + (2yz – 4z) k curl G G ? = (c) H = 5yzi+ (z² – 5xz) j + (5xy + 2yz) k: curl H H? =
3. Find a vector potential field for Ġ = [6xz + x³, –3x?y – y?, 4x + 2zy – 3z2], i.e. find F such that V x F = Ġ.
7. Find the outward flux of the field F(x, y) = -yi + x² j across the closed counter-clockwise curve consisting of the top semi-circle of + y? = 4 followed by the straight line from (-2,0) to (2,0).
Decide if it is gradient field. Justify.

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Suppose S is the union of the cylinder x² + y² = 1 for 0 ≤ z ≤ 1 and the disk x² + y² ≤ 1 at z = 1. Suppose F is a vector field such that ▼ × F = ( sinh(z)(x² + y²), ze²v+cos(x+v), (x2 + y) tan−¹(2)). Calculate the flux of V x F though S.