Answered: Surface integrals of vector fields Find…

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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Question

Surface integrals of vector fields Find the flux of the following vector field across the given surface with the specified orientation. You may use either an explicit or a parametric description of the surface.

F = ⟨x, y, z⟩ across the slanted surface of the cone z² = x² + y²,
for 0 ≤ z ≤ 1; normal vectors point upward.

Expert Solution

Step 1

Given:

$F = <x, y, z>$ is a vector field that fills the cone's surface $z^{2} = x^{2} + y^{2}$ from $0 \leq z \leq 1$ in an upward direction.

Step 2

Formula:

Surface Integral of a Vector Field :

$\iint_{S} F \cdot ndS = \iint_{R} F \cdot (t_{u} \times t_{v}) dA$ , where $t_{u} = \frac{\partial r}{\partial u} and t_{v} = \frac{\partial r}{\partial v}$

Solution:

Here the surface $z^{2} = x^{2} + y^{2}$ ,

Differentiate z with respect to y to obtain $z_{x}$ ,

$2 zdz = 2 xdx \frac{d z}{d x} = \frac{x}{z} z_{x} = \frac{x}{z}$

Differentiate z with respect to y to obtain $z_{y}$ ,

$2 zdz = 2 xdx \frac{d z}{d y} = \frac{y}{z} z_{y} = \frac{y}{z}$

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