Hi, I wonder if i can get help to solve this problem.Thanks, Anna-Sara yzFind the flux of the field Fup through the part of the1+ xy,paraboloid z =1-x² – y? that resides above the ry plane.%3D

Given field F=yzxz1+xy The part of the paraboloid z=1-x2-y2 cut off by the xy-plane with upward.…

Answered: yz Find the flux of the field F up…

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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Hi,

I wonder if i can get help to solve this problem.

Thanks,

Anna-Sara

yz
Find the flux of the field F
up through the part of the
1+ xy,
paraboloid z =1-x² – y? that resides above the ry plane.
%3D

Expert Solution

Step 1

Given field $F = (\begin{matrix} y z \\ x z \\ 1 + x y \end{matrix})$

The part of the paraboloid $z = 1 - x^{2} - y^{2}$ cut off by the $x y -$ plane with upward.

Recollect the Stokes theorem

$\int_{c} F . d r = \iint_{c} c u r l F n d s$

$c u r l F = |\begin{matrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ y z & x z & 1 + x y \end{matrix}| = i (\frac{\partial (1 + x y)}{\partial y} - \frac{\partial (x z)}{\partial z}) - j (\frac{\partial (1 + x y)}{\partial x} - \frac{\partial (y z)}{\partial z}) + k (\frac{\partial (x z)}{\partial x} - \frac{\partial (y z)}{\partial y})$

$= i (x - x) - j (y - y) + k (z - z) = 0 i - 0 j + 0 k$

$= <0, 0, 0>$

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