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Home Vector Calculus Theorems of Stokes and Gauss Problems  
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In Problems 16, find the curl and divergence of the vector field. 7 Prove that for every vector field F(x, y, z) with continuous second partials, div(curl F) = 0. 8 Given a function f(x, y, z) with continuous second partials, show that 9 Use Stokes' Theorem to evaluate the surface integral where S is the portion of the paraboloid z = 1  x^{2}  y^{2} above the (x, y) plane and F(x, y, z) = xy^{2}i  x^{2}yj + xyzk. (S is oriented with the top side positive.) 10 Use Stokes' Theorem to evaluate the line integral where S is the portion of the plane z = 2x + 5y inside the cylinder x^{2} + y^{2} = 1 oriented with the top side positive. 11 Use Stokes' Theorem to evaluate the line integral where S is the portion of the plane z = px + qy + r over a region D of area A, oriented with the top side positive. 12 Use Stokes' Theorem to show that the line integral for any oriented surface S. 13 Use Gauss' Theorem to compute the surface integral where E is the rectangular box 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c. 14 Use Gauss' Theorem to compute the surface integral where E is the rectangular box 0 ≤ x ≤ 1l, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1. 15 Use Gauss' Theorem to evaluate where E is the region 0 ≤ x ≤ 1, 0 ≤ y ≤ x, 0 ≤ z ≤ x + y. 16 Use Gauss' Theorem to evaluate where E is the sphere x^{2} + y^{2} + z^{2} ≤ 4. 17 Use Gauss' Theorem to evaluate where E is the hemisphere 0 ≤ z ≤ √(l  x^{2}  y^{2}). 18 Use Gauss' Theorem to evaluate where S is the cylinder x^{2} + y^{2} ≤ 1, 0 ≤ z ≤ 4. 19 Use Gauss' Theorem to evaluate where E is the part of the cone above the (x, y) plane.


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