The divergence theorem practice problems
Flux out of a closed surface as the triple integral of the divergence. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(y z\right)\mathbf i + \left(- 2 y\right)\mathbf j + \left(z^{2}\right)\mathbf k \) across the surface of the box \( \displaystyle [0, 3] \times [0, 2] \times [0, 1] \).
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(0\right)\mathbf i + \left(y^{2}\right)\mathbf j + \left(z^{2}\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 4 \).
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(0\right)\mathbf i + \left(y^{2}\right)\mathbf j + \left(0\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 1 \).
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(0\right)\mathbf i + \left(y^{2}\right)\mathbf j + \left(0\right)\mathbf k \) across the surface of the box \( \displaystyle [0, 2] \times [0, 1] \times [0, 3] \).
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(x\right)\mathbf i + \left(- 2 y\right)\mathbf j + \left(z^{3}\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 4 \).
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(x^{2}\right)\mathbf i + \left(y\right)\mathbf j + \left(x y\right)\mathbf k \) across the surface of the box \( \displaystyle [0, 3] \times [0, 1] \times [0, 1] \).
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(x^{3}\right)\mathbf i + \left(y\right)\mathbf j + \left(z^{2}\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 1 \).
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(x^{2}\right)\mathbf i + \left(y^{3}\right)\mathbf j + \left(z^{3}\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 1 \).
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(x^{3}\right)\mathbf i + \left(y^{3}\right)\mathbf j + \left(x y\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 4 \).
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(y z\right)\mathbf i + \left(y\right)\mathbf j + \left(z^{2}\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 1 \).