I have these three problems and tried to solve them....
Here are my works I'm wondering if they are correct....
Thank you!
1. A fluid with density 1,330 flows with a velocity field v = y i + j + z k. Find the rate of flow upward through the paraboloid:
. . . . .\(\displaystyle z\, =\, 9\, -\, \dfrac{1}{3}\, (x^2\, +\, y^2),\, \mbox{ where }\, x^2\, +\, y^2\, \leq\, 36\)
3. Use the Divergence Theorem the calculate the surface integral:
. . . . .\(\displaystyle \iint_S\, \mathbf{F}\, d\mathbf{S},\, \mbox{ where }\, \mathbf{F}(x,\, y,\, z)\, =\, xy\mathbf{i}\, +\, yz\mathbf{j}\, +\, zx\mathbf{k}\)
. . . . .\(\displaystyle \mbox{and }\, S\, \mbox{ is the surface of the solid cylinder }\, x^2\, +\, y^2\, \leq\, 9,\, 0\, \leq\, z\, \leq\, 2.\)
7. Verify the Divergence Theorem for E : x2 + y2 + z2 = a2 and the field F = x i + y j + z k.
Here are my works I'm wondering if they are correct....
Thank you!
1. A fluid with density 1,330 flows with a velocity field v = y i + j + z k. Find the rate of flow upward through the paraboloid:
. . . . .\(\displaystyle z\, =\, 9\, -\, \dfrac{1}{3}\, (x^2\, +\, y^2),\, \mbox{ where }\, x^2\, +\, y^2\, \leq\, 36\)
3. Use the Divergence Theorem the calculate the surface integral:
. . . . .\(\displaystyle \iint_S\, \mathbf{F}\, d\mathbf{S},\, \mbox{ where }\, \mathbf{F}(x,\, y,\, z)\, =\, xy\mathbf{i}\, +\, yz\mathbf{j}\, +\, zx\mathbf{k}\)
. . . . .\(\displaystyle \mbox{and }\, S\, \mbox{ is the surface of the solid cylinder }\, x^2\, +\, y^2\, \leq\, 9,\, 0\, \leq\, z\, \leq\, 2.\)
7. Verify the Divergence Theorem for E : x2 + y2 + z2 = a2 and the field F = x i + y j + z k.
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