Verify the divergence theorem using the vector filed...

Question:

Verify the divergence theorem using the vector filed {eq}\displaystyle \vec{F} = 4x\vec{i} - 2y^{2}\vec{j} + z^{2}\vec{k}, {/eq} if the enclosed surface is the cylinder {eq}\displaystyle x^{2} + y^{2} = 4 {/eq} between {eq}z = 0 {/eq} and {eq}z = 3. {/eq}

Divergence Thorem:

The divergence theorem has the following formula {eq}\displaystyle \int \int _S F\cdot dS =\int \int \int _E div\:F\left ( x,y,z \right )dV {/eq} where {eq}F {/eq} is the vector field {eq}\displaystyle F=Pi+Qj+Rk {/eq} and {eq}\displaystyle div\:F {/eq} means the divergence of {eq}F {/eq} which has the formula {eq}\displaystyle div\:F=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z} {/eq}

Answer and Explanation: 1

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First let us solve for {eq}div\:F {/eq}

{eq}\displaystyle div\:F=\frac{\partial 4x}{\partial x}+\frac{\partial (-2y^{2})}{\partial y}+\frac{\partial...

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Divergence Theorem: Definition, Applications & Examples
Divergence Theorem: Definition, Applications & Examples

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Chapter 26 / Lesson 18
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Learn the divergence theorem formula. Explore examples of the divergence theorem. Understand how to measure vector surface integrals and volume integrals.


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