Please show all work **Problem 3:** Find $ \text{curl} \, \vec{F} $ for $ \vec{F} = xyz^2 \vec{i} + x^4 yz^2 \vec{j} + x^4 y^2 z \vec{k} $.**Answer:** $ \text{curl} \, \vec{F} = (2xyz - 4x^3 y^2 z) \vec{j} + (4x^3 yz^2 - xz^2) \vec{k} $.

Given, F→=xyz2i→+x4yz2j→+x4y2zk→ Used formulas, ∂∂xxn=nxn-1

Answered: Find curlF for F = xyz?i+ x*yz?j+…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Concept explainers

Question

Please show all work

**Problem 3:** Find $ \text{curl} \, \vec{F} $ for $ \vec{F} = xyz^2 \vec{i} + x^4 yz^2 \vec{j} + x^4 y^2 z \vec{k} $.

**Answer:** $ \text{curl} \, \vec{F} = (2xyz - 4x^3 y^2 z) \vec{j} + (4x^3 yz^2 - xz^2) \vec{k} $.

Expert Solution

Step 1

Given,

$\vec{F} = x y z^{2} \vec{i} + x^{4} y z^{2} \vec{j} + x^{4} y^{2} z \vec{k}$

Used formulas,

$\frac{\partial}{\partial x} x^{n} = n x^{n - 1}$

Step 2

$\begin{array}{rcl} c u r l \vec{F} & = & \nabla \times \vec{F} \\ = & |\begin{matrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ x y z^{2} & x^{4} y z^{2} & x^{4} y^{2} z \end{matrix}| \\ = & (\frac{\partial}{\partial y} (x^{4} y^{2} z) - \frac{\partial}{\partial z} (x^{4} y z^{2})) \vec{i} - (\frac{\partial}{\partial x} (x^{4} y^{2} z) - \frac{\partial}{\partial z} (x y z^{2})) \vec{j} + (\frac{\partial}{\partial x} (x^{4} y z^{2}) - \frac{\partial}{\partial y} (x y z^{2})) \vec{k} \\ = & (x^{4} z \frac{\partial}{\partial y} (y^{2}) - x^{4} y \frac{\partial}{\partial z} (z^{2})) \vec{i} - (y^{2} z \frac{\partial}{\partial x} (x^{4}) - x y \frac{\partial}{\partial z} (z^{2})) \vec{j} + (y z^{2} \frac{\partial}{\partial x} (x^{4}) - x z^{2} \frac{\partial}{\partial y} (y)) \vec{k} \\ = & (x^{4} z (2 y) - x^{4} y (2 z)) \vec{i} - (y^{2} z (4 x^{3}) - x y (2 z)) \vec{j} + (y z^{2} (4 x^{3}) - x z^{2} (1)) \vec{k} \\ = & (2 x^{4} y z - 2 x^{4} y z) \vec{i} - (4 x^{3} y^{2} z - 2 x y z) \vec{j} + (4 x^{3} y z^{2} - x z^{2}) \vec{k} \\ = & 0 \vec{i} - (4 x^{3} y^{2} z - 2 x y z) \vec{j} + (4 x^{3} y z^{2} - x z^{2}) \vec{k} \\ = & - (4 x^{3} y^{2} z - 2 x y z) \vec{j} + (4 x^{3} y z^{2} - x z^{2}) \vec{k} \\ = & (2 x y z - 4 x^{3} y^{2} z) \vec{j} + (4 x^{3} y z^{2} - x z^{2}) \vec{k} \end{array}$

Step by step

Solved in 3 steps

Knowledge Booster

$Matrix Operations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions