Use the Divergence Theorem to evaluate \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\), where \(\mathbf{F}(x, y, z)=z^{2} x \mathbf{i}+\left(\frac{y^{3}}{3}+\cos z\right) \mathbf{j}+\left(x^{2} z+y^{2}\right) \mathbf{k}\) and \(S\) is the top half of the sphere \(x^{2}+y^{2}+z^{2}=4\). (Hint: Note that \(S\) is not a closed surface. First compute integrals over \(S_{1}\) and \(S_{2}\), where \(S_{1}\) is the disk \(x^{2}+y^{2} \leq 4\), oriented downward, and \(S_{2}=S_{1} \cup S\).)
Help Entering Answers (1 point) Use the Divergence Theorem to evaluate F . dS where F =くz2xHFz, y + 2 tan(2), X22-1 and S is the top half of the sphere x2 +y2 25 Hint: S is not a closed surface. First compute integrals over S and S2, where Si is the disk x2 +y s 25, z 0 oriented downward and s,-sus, F-ds, = 滋 dy dx F.dS2 = S2 where X1 = 리= Z2 = IE F-ds, =...
Use the Divergence Theorem to evaluate ∫∫S F·dS, where F(x,y,z)=z²xi+(y³/3+sin z) j+(x² z+y²) k and S is the top half of the sphere x²+y²+z²=4 . (Hint: Note that S is not a closed surface, First compute integrals over S₁ and S₂, where S₁ is the disk x²+y² ≤ 4, oriented downward, and S₂=S₁ ∪ S.)
Use the Divergence Theorem to evaluate ∬SF⋅dS∬SF⋅dS where F=〈z2x,y33+3tan(z),x2z−1〉F=〈z2x,y33+3tan(z),x2z−1〉 and SS is the top half of the sphere x2+y2+z2=9x2+y2+z2=9. (1 point) Use the Divergence Theorem to evaluate FdS where F2x +3 tan2).^z-1 and S is the top half of the sphere x2 +y2 + z2 -9 Hint: S is not a closed surface. First compute integrals overs, and S2 , where S, is the disk x2 + y2 < 9, z = 0 oriented downward and S2 = S U...
Use the divergence theorem to calculate the flux of the vector field \(\vec{F}(x, y, z)=x^{3} \vec{i}+y^{3} \vec{j}+z^{3} \vec{k}\) out of the closed, outward-oriented surface \(S\) bounding the solid \(x^{2}+y^{2} \leq 16,0 \leq z \leq 3\).
Evaluate the surface integral \(\iint_{S} F \cdot d \mathbf{S}\) for the given vector field \(\mathbf{F}\) and the oriented surface \(S\). In other words, find the flux of \(F\) across 5 . For closed surfaces, use the positive (outward) orientation.$$ \mathbf{F}(x, y, z)=x \mathbf{i}+3 y \mathbf{j}+2 z \mathbf{k} $$\(S\) is the cube with vertices \((\pm 1, \pm 1, \pm 1)\)
Use a parametrization to find the flux\(\iint_{S} \mathbf{F} \cdot \mathbf{n} \mathrm{d} \sigma\)of the field \(\mathbf{F}=\frac{9 x \mathbf{i}+9 y \mathbf{j}+9 z \mathbf{k}}{\sqrt{x^{2}+y^{2}+z^{2}}}\) across the portion of the sphere \(x^{2}+y^{2}+z^{2}=25\) in the first octant in the direction away from the origin.The flux is _______
Use Stokes's theorem to calculate \(\iint_{S}(\nabla \times \boldsymbol{F}) \cdot d S\) where$$ \boldsymbol{F}=(x-z) \boldsymbol{i}+\left(x^{3}+y z\right) \boldsymbol{j}-3 x y^{2} \boldsymbol{k} $$and \(S\) is the surface of the cone \(z=2-\sqrt{x^{2}+y^{2}}\) above the \(x y\) plane.
Evaluate \(\int_{C} \mathrm{~F} \cdot d \mathbf{r}\) using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results.$$ \int_{C}[4(2 x+7 y) \mathbf{i}+14(2 x+7 y) \mathbf{j}] \cdot d \mathbf{r} $$C: smooth curve from \((-7,2)\) to \((3,2)\)Evaluate \(\int_{C} \mathrm{~F} \cdot d \mathbf{r}\) using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results.$$ \int_{C} \cos (x) \sin (y) d x+\sin (x) \cos (y) d y $$C: line segment from \((0,-\pi)\) to \(\left(\frac{3 \pi}{2},...
Provide correct answer Use the Divergence Theorem to evaluate //F. ds where F = (4x", 4y?, 17) and S is the sphere x² + y2 + z = 25 oriented by the outward normal. The surface integral equals
Question 1. Determine whether or not \(\mathrm{F}(x, y)=e^{x} \sin y \mathbf{i}+e^{x} \cos y_{\mathbf{j}}\) is a conservative field. If it is, find its potential function \(f\).Question 2. Find the curl and the divergence of the vector field \(\mathbf{F}=\sin y z \mathbf{i}+\sin z x \mathbf{j}+\sin x y \mathbf{k}\)Question 3. Find the flux of the vector field \(\mathbf{F}=z \mathbf{i}+y \mathbf{j}+x \mathbf{k}\) across the surface \(r(u, v)=\langle u \cos v, u \sin v, v\rangle, 0 \leq u \leq 1,0 \leq v \leq \pi\) with...