Solution: In Exercises 718, use the Divergence Theorem to evaluate SF N dS and find the

In Exercises 1-6, verify the Divergence Theorem by evaluating \(\int_{S} \int F \cdot N d S\) as a surface integral and as a triple integral. \(\mathbf{F}(x, y, z)=2 x \mathbf{i}-2 y \mathbf{j}+z^{2} \mathbf{k}\) S: cube bounded by the planes x = 0, x = a, y = 0, y = a, z = 0, z = a Text Transcription: int_S int F cdot N dS F(x,y,z)=2xi-2yj+z^2k

In Exercises 1-6, verify the Divergence Theorem by evaluating \(\int_{S} \int F \cdot N d S\) as a surface integral and as a triple integral. \(\mathbf{F}(x, y, z)=2 x \mathbf{i}-2 y \mathbf{j}+z^{2} \mathbf{k}\) S: cylinder \(x^{2}+y^{2}=4, \quad 0 \leq z \leq h\) Text Transcription: int_S int F cdot N dS x^2+y^2=4, 0 <= z <= h

In Exercises 1-6, verify the Divergence Theorem by evaluating \(\int_{S} \int F \cdot N d S\) as a surface integral and as a triple integral. F(x, y, z) = (2x - y)i - (2y - z)j + zk S: surface bounded by the plane 2x + 4y + 2z = 12 and the coordinate planes Text Transcription: int_S int F cdot N dS

In Exercises 1-6, verify the Divergence Theorem by evaluating \(\int_{S} \int F \cdot N d S\) as a surface integral and as a triple integral. F(x, y, z) = xyi + zj + (x + y)k S: surface bounded by the planes y = 4 and z = 4 - x and coordinate planes Text Transcription: int_S int F cdot N dS

In Exercises 1-6, verify the Divergence Theorem by evaluating \(\int_{S} \int F \cdot N d S\) as a surface integral and as a triple integral. \(\mathbf{F}(x, y, z)=x z \mathbf{i}+z y \mathbf{j}+2 z^{2} \mathbf{k}\) S : surface bounded by \(z=1-x^{2}-y^{2}\) and z = 0 Text Transcription: int_S int F cdot N dS F(x,y,z)=xzi+zyj+2z^2k z=1-x^2-y^2

In Exercises 1-6, verify the Divergence Theorem by evaluating \(\int_{S} \int F \cdot N d S\) as a surface integral and as a triple integral. \(\mathbf{F}(x, y, z)=x y^{2} \mathbf{i}+y x^{2} \mathbf{j}+e \mathbf{k}\) S : surface bounded by \(z=\sqrt{x^{2}+y^{2}}\) and z = 4 Text Transcription: int_S int F cdot N dS F(x,y,z)=xy^2i+yx^2j+ek z=sqrt x^2+y^2

In Exercises 7-18, use the Divergence Theorem to evaluate \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. \(\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}\) S: x = 0, x = a, y = 0, y = a, z = 0, z = a Text Transcription: int_S int F cdot N DS F(x,y,z)=x^2i+y^2j+z^2k

In Exercises 7-18, use the Divergence Theorem to evaluate \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. \(\mathbf{F}(x, y, z)=x^{2} z^{2} \mathbf{i}-2 y \mathbf{j}+3 x y z \mathbf{k}\) S: x = 0, x = a, y = 0, y = a, z = 0, z = a Text Transcription: int_S int F cdot N DS F(x,y,z)=x^2z^2i-2yj+3xyzk

In Exercises 7-18, use the Divergence Theorem to evaluate \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. \(\mathbf{F}(x, y, z)=x^{2} \mathbf{i}-2 x y \mathbf{j}+x y z^{2} \mathbf{k}\) \(S: \quad z=\sqrt{a^{2}-x^{2}-y^{2}}, z=0\) Text Transcription: int_S int F cdot N DS F(x,y,z)=x^2i-2xyj+xyz^2k S: z=sqrt a^2-x^2-y^2, z=0

In Exercises 7-18, use the Divergence Theorem to evaluate \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x, y ,z) = xyi + yzj - yzk \(S: z=\sqrt{a^{2}-x^{2}-y^{2}}, z=0\) Text Transcription: int_S int F cdot N DS S: z=sqrt a^2-x^2-y^2, z=0

In Exercises 7-18, use the Divergence Theorem to evaluate \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x, y ,z) = xi + yj - zk \(S: x^{2}+y^{2}+z^{2}=9\) Text Transcription: int_S int F cdot N DS S: x^2+y^2+z^2=9

In Exercises 7-18, use the Divergence Theorem to evaluate \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x,y,z) = xyzj \(S: x^{2}+y^{2}=4\), z = 0, z = 5 Text Transcription: int_S int F cdot N DS S: x^2+y^2=4

In Exercises 7-18, use the Divergence Theorem to evaluate \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y^{2} \mathbf{j}-z \mathbf{k}\) \(S: x^{2}+y^{2}=25\), z = 0, z = 7 Text Transcription: int_S int F cdot N DS F(x,y,z)=xi+y^2j-zk S:x^2+y^2=25

In Exercises 7-18, use the Divergence Theorem to evaluate \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. \(\mathbf{F}(x, y, z)=\left(x y^{2}+\cos z\right) \mathbf{i}+\left(x^{2} y+\sin z\right) \mathbf{j}+e^{z} \mathbf{k}\) \(S: z=\frac{1}{2} \sqrt{x^{2}+y^{2}}\), z = 8 Text Transcription: int_S int F cdot N DS F(x,y,z)=(xy^2+cos z)i+(x^2y+sin z)j+e^zk S:z=1/2 sqrt x^2+y^2

In Exercises 7-18, use the Divergence Theorem to evaluate \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. \(\mathbf{F}(x, y, z)=x^{3} \mathbf{i}+x^{2} y \mathbf{j}+x^{2} e^{y} \mathbf{k}\) S: z = 4 - y, z = 0, x = 0, x = 6, y = 0 Text Transcription: int_S int F cdot N DS F(x,y,z)=x^3i+x^2yj+x^2e^yk

In Exercises 7-18, use the Divergence Theorem to evaluate \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. \(\mathbf{F}(x, y, z)=x e^{z} \mathbf{i}+y e^{z} \mathbf{j}+e^{z} \mathbf{k}\) S: z = 4 - y, z = 0, x = 0, x = 6, y = 0 Text Transcription: int_S int F cdot N DS F(x,y,z)=xe^zi+ye^zj+e^zk

In Exercises 7-18, use the Divergence Theorem to evaluate \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x, y, z) = xyi + 4yj + xzk \(S: x^{2}+y^{2}+z^{2}=16\) Text Transcription: int_S int F cdot N DS S: x^2+y^2+z^2=16

In Exercises 7-18, use the Divergence Theorem to evaluate \(\int_{S} \int \mathrm{F} \cdot \mathrm{N} d S\) and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x, y, z) = 2(xi + yj + zk) \(S: z=\sqrt{4-x^{2}-y^{2}}\), z = 0 Text Transcription: int_S int F cdot N DS S: z=sqrt 4-x^2-y^2

In Exercises 19 and 20, evaluate \(\int_{S} \int \operatorname{curl} F \cdot N d S\) where S is the closed surface of the solid bounded by the graphs of x = 4 and \(z=9-y^{2}\), and the coordinate planes. \(\mathbf{F}(x, y, z)=\left(4 x y+z^{2}\right) \mathbf{i}+\left(2 x^{2}+6 y z\right) \mathbf{j}+2 x z \mathbf{k}\) Text Transcription: int_S int curl F cdot N dS z=9-y^{2} F(x, y, z)=(4xy+z^2) i+(2x^2+6yz)j+2xzk

In Exercises 19 and 20, evaluate \(\int_{S} \int \operatorname{curl} F \cdot N d S\) where S is the closed surface of the solid bounded by the graphs of x = 4 and \(z=9-y^{2}\), and the coordinate planes. F(x, y, z) = xy cos zi + yz sin xj + xyzk Text Transcription: int_S int curl F cdot N dS z=9-y^2

State the Divergence Theorem.

How do you determine if a point \(\left(x_{0}, y_{0}, z_{0}\right)\) in a vector field is a source, a sink, or incompressible? Text Transcription: (x_0,y_0,z_0)

(a) Use the Divergence Theorem to verify that the volume of the solid bounded by a surface S is \(\int_{S} \int x d y d z=\int_{S} \int y d z d x=\int_{S} \int z d x d y\). (b) Verify the result of part (a) for the cube bounded by x = 0, x = a, y = 0, y = a, z = 0, and z = a. Text Transcription: int_S int x dy dz=int_S int y dz dx=int_S int z dx dy

Let F(x, y, z) = xi + yj + zk and let S be the cube bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0, and z = 1. Verify the Divergence Theorem by evaluating \(\int_{S} \int \mathbf{F} \cdot \mathbf{N} d S\) as a surface integral and as a triple integral. Text Transcription: int_S int F cdot N dS

Verify that \(\int_{S} \int \operatorname{curl} \mathbf{F} \cdot \mathbf{N} d S=0\) for any closed surface S. Text Transcription: int_S int curl F cdot N dS=0

For the constant vector field \(\mathbf{F}(x, y, z)=a_{1} \mathbf{i}+a_{2} \mathbf{j}+a_{3} \mathbf{k}\), verify that \(\int_{S} \int \mathbf{F} \cdot \mathbf{N} d S=0\) where V is the volume of the solid bounded by the closed surface S. Text Transcription: F(x,y,z)=a_1i+a_2j+a_3k int_S int F cdot N dS=0

Given the vector field F(x, y, z) = xi + yj + zk, verify that \(\int_{S} \int \mathbf{F} \cdot \mathbf{N} d S=3 V\) where V is the volume of the solid bounded by the closed surface S. Text Transcription: int_S int F cdot N dS=3V

Given the vector field F(x, y, z) = xi + yj + zk, verify that \(\frac{1}{\|\mathbf{F}\|} \int_{S} \int \mathbf{F} \cdot \mathbf{N} d S=\frac{3}{\|\mathbf{F}\|} \iiint_{Q} d V\). Text Transcription: 1/||F|| int_S int F cdot N dS= 3/||F|| iiint_Q dV

In Exercises 29 and 30, prove the identity, assuming that Q, S, and N meet the conditions of the Divergence Theorem and that the required partial derivatives of the scalar functions f and g are continuous. The expressions \(D_{\mathrm N} f\) and \(D_{\mathrm N} f\) are the derivatives in the direction of the vector N and are defined by \(D_{\mathrm{N}} f=\nabla f \cdot \mathrm{N}, \quad D_{\mathrm{N}} g=\nabla g \cdot \mathrm{N}\). \(\iint_{Q} \int\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V=\int_{S} \int f D_{\mathbf{N}} g d S\) [Hint: Use div \((f \mathbf{G})=f \operatorname{div} \mathbf{G}+\nabla f \cdot \mathbf{G}\).] Text Transcription: D_Nf D_Ng D_Nf=nabla f cdot N, N_Ng=nabla g cdot N iiint_Q(f nabla^2 g+nabla f cdot nabla g)dV= int_S int f D_Ng dS (fG)=f div G+nabla f cdot G

In Exercises 29 and 30, prove the identity, assuming that Q, S, and N meet the conditions of the Divergence Theorem and that the required partial derivatives of the scalar functions f and g are continuous. The expressions \(D_{\mathrm N} f\) and \(D_{\mathrm N} f\) are the derivatives in the direction of the vector N and are defined by \(D_{\mathrm{N}} f=\nabla f \cdot \mathrm{N}, \quad D_{\mathrm{N}} g=\nabla g \cdot \mathrm{N}\). \(\iiint_{Q}\left(f \nabla^{2} g-g \nabla^{2} f\right) d V=\int_{S} \int\left(f D_{\mathbf{N}} g-g D_{\mathbf{N}} f\right) d S\) (Hint: Use Exercise 29 twice.) Text Transcription: D_Nf D_Ng D_Nf=nabla f cdot N, N_Ng=nabla g cdot N iiint_Q (f nabla^2g-g nabla^2 f) dV=int_S int(f D_N g-g D_N f) dS