Let \vec{F}(x, y, z) = -4z \vec{i} + 3x \vec{k} and S be the closed square pyramid with height 14...

Question:

Let {eq}\vec{F}(x, y, z) = -4z \vec{i} + 3x \vec{k} {/eq} and {eq}S {/eq} be the closed square pyramid with height {eq}14 {/eq} and base on the {eq}xy {/eq}-plane with side length {eq}3 {/eq}. The pyramid {eq}S {/eq} is oriented outward. Compute the flux of {eq}\vec{F} {/eq} through {eq}S {/eq}.

Computing the Flux:

For computing the flux, we have to use the Gauss Divergence theorem.

The general form of the Gauss Divergence theorem is {eq}\iint_{S} \vec{F} \cdot d\vec{A} = \iiint_{V} div F \ dV {/eq}

By applying the given function in the general form.

Then evaluate the function to get a resultant part.

Answer and Explanation: 1

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The given vector field is {eq}\vec{F}\left ( x, y, z \right ) = -4z \vec{i} + 3x \vec{k} {/eq}.

Where {eq}S {/eq} be the closed square pyramid.

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Evaluating Definite Integrals Using the Fundamental Theorem

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Chapter 16 / Lesson 2
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The fundamental theorem of calculus makes finding your definite integral almost a piece of cake. See how the definite integral becomes a subtraction problem after applying the fundamental theorem of calculus.


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