find the curl and the divergence of the givenvector field. 15. F(x, y, z) = xye*i − x³yze²j + xy²e³k

Answered: 15. F(x, y, z) = xye¹i — xªyze¹j +…

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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Related questions

Question

find the curl and the divergence of the given
vector field.

15. F(x, y, z) = xye*i − x³yze²j + xy²e³k

Expert Solution

Step 1

Given vector field is

$F (x, y, z) = x y e^{x} i - x^{3} y z e^{z} j + x y^{2} e^{y} k$

Here

$C u r l F = |\begin{matrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ x y e^{x} & - x^{3} y z e^{z} & x y^{2} e^{y} \end{matrix}| = (\frac{\partial}{\partial y} (x y^{2} e^{y}) - \frac{\partial}{\partial z} (- x^{3} y z e^{z})) i + (\frac{\partial}{\partial z} (x y e^{x}) - \frac{\partial}{\partial x} (x y^{2} e^{y})) j + (\frac{\partial}{\partial x} (- x^{3} y z e^{z}) - \frac{\partial}{\partial y} (x y e^{x})) z = (x y (y + 2) e^{y} + x^{3} y (z + 1) e^{z}) i + (0 - y^{2} e^{y}) j + (- 3 x^{2} y z e^{z} - x e^{x}) k$