Mathematics: Verify that the divergence theorem is true for the vector field f on the region e. give the flux. f(x, y, z) = 4xi + xyj + 5xzk, e is the cube bounded by the planes x = 0, x = 2, y = 0, y = 2, z = 0, and z = 2.

Verify that the divergence theorem is true for the

Ответ:

TYUqnn2077

12.03.2022

By the divergence theorem, the flux of $\mathbf f(x,y,z)=4x\,\mathbf i+xy\,\mathbf j+5xz\,\mathbf k$ across the boundary $\partial E$ of the cube is equal to the integral of $\nabla\cdot\mathbf f$ over :

$\displaystyle\iint_{\partial E}\mathbf f\cdot\mathrm d\mathbf S=\iiint_E\nabla\cdot\mathbf f\,\mathrm dV$

We have divergence

$\nabla\cdot\mathbf f=\dfrac{\partial(4x)}{\partial x}+\dfrac{\partial(xy)}{\partial y}+\dfrac{\partial(5xz)}{\partial z}=4+6x$

and so the flux is

$\displaystyle\iiint_E(4+6x)\,\mathrm dV=\int_{z=0}^{z=2}\int_{y=0}^{y=2}\int_{x=0}^{x=2}(4+6x)\,\mathrm dx\,\mathrm dy\,\mathrm dz$

$=4(4x+3x^2)\bigg|_{x=0}^{x=2}=80$

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Ответ:

payshencec21

10.01.2020

Mathematics

Answer

c) carter

Explanation

Remember that A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed number called the ratio. So to prove if a sequence is geometric, y we just need to verify that the ratio between any tow terms of the sequence is the same.

Now, to find the ratio of a geometric sequence, we use the formula:

$r=\frac{a_{n}}{a_{n-1}}$