- #1
Saladsamurai
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- 7
Homework Statement
Evaluate div v at P = (0, 0, 0) by actually evaluating [itex](\int_S\mathbf{\hat{n}}\cdot \mathbf{v}\,dA)/V[/itex] and taking the limit as B-->0. Take B to be the cube [itex]|x|\le\epsilon,|y|\le\epsilon,|z|\le\epsilon[/itex]. Let [itex]\mathbf{v} = x\mathbf{\hat{i}} + 2y\mathbf{\hat{j}} - 4z^3\mathbf{\hat{k}}[/itex]
Homework Equations
The Attempt at a Solution
So what I need to do is to first find the integral [itex]\int_S\mathbf{\hat{n}}\cdot \mathbf{v}\,dA[/itex] and to do so, I will break it up into 6 integrals, one for each face of the cube.First I have a question: the way that the bounds of the cube are given suggest that the cube is [itex]2\epsilon[/itex] in length in each direction. I am wondering how I am to position my coordinate system. Should it be centered in the cube? Should it be at a corner? Does it matter? I would like to think that it does not matter, but I cannot figure out how to justify that assumption.
I have more questions, but I would like to clarify this one first. I started the problem by positioning the origin at the center of the cube, but I want to confirm that's ok before typing my work in.
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