11.7: Stokes' Theorem
- Page ID
- 144364
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- Use Stokes' theorem to evaluate \(\displaystyle \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}\), where \(\mathbf{F}(x, y, z) = xy\mathbf{i} - z\mathbf{j}\), and where \(S\) is the surface of the unit cube \([0, 1] \times [0, 1] \times [0, 1]\) with the bottom face \(z = 0\) removed and normals oriented outward.
- Use Stokes' theorem to evaluate \(\displaystyle \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}\), where \(\mathbf{F}(x, y, z) = xy\mathbf{i} + x^2\mathbf{j} + z^2\mathbf{k}\), and where \(S\) is the portion of the surface \(z = x^2 + y^2\) with \(z \leq y\) and normals oriented downward.
- Use Stokes' theorem to evaluate \(\displaystyle \int_C \mathbf{F} \cdot d\mathbf{r}\), where \(\mathbf{F}(x, y, z) = y^2\mathbf{i} + z^2\mathbf{j} + x^2\mathbf{k}\), and where \(C\) is the boundary of the plane \(x + y + z = 1\) restricted in to the first octant oriented counterclockwise.