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Let 02 be the volume in the first octant bounded by the planes 0, " = 0 and Ity+z= 1 aud oriented with the outward UIL normal. Let $ be that part of €02 ...

Question

Let 02 be the volume in the first octant bounded by the planes 0, " = 0 and Ity+z= 1 aud oriented with the outward UIL normal. Let $ be that part of €02 that comes from removing the piece that lies in the plane and between the lines 0. y 0, and (So $ consists of three triangular faces ) Let G (0.0.1).Evaluate Js G ndS by using the divergence theorem trade three surface integrals for one Evaluate Js G -ndS by using Stokes" theorem_ HINT: Find vector field whose curl is G_ Since G has

Let 02 be the volume in the first octant bounded by the planes 0, " = 0 and Ity+z= 1 aud oriented with the outward UIL normal. Let $ be that part of €02 that comes from removing the piece that lies in the plane and between the lines 0. y 0, and (So $ consists of three triangular faces ) Let G (0.0.1). Evaluate Js G ndS by using the divergence theorem trade three surface integrals for one Evaluate Js G -ndS by using Stokes" theorem_ HINT: Find vector field whose curl is G_ Since G has only K-component look fr vector field of the form F P(x.y)i- Q(I.y)j



Answers

Let $\sigma$ be the surface of the solid $G$ that is enclosed by the paraboloid $z=1-x^{2}-y^{2}$ and the plane $z=0 .$ Use a CAS to verify Formula ( 1) in the Divergence Theorem for the vector field $$ \mathbf{F}=\left(x^{2} y-z^{2}\right) \mathbf{i}+\left(y^{3}-x\right) \mathbf{j}+(2 x+3 z-1) \mathbf{k} $$ by evaluating the surface integral and the triple integral.

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