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Calculus 3 : Line Integrals

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Example Questions

Example Question #1 : Divergence

Compute $div\vec{F}$ , where $\vec{F}=10x^2\ln(yz)\vec{i}+xyz\vec{j}+xz\ln(y)\vec{k}$

Possible Answers:

$div\vec{F}=0$

$div\vec{F}=20x\ln(yz)-xz+x\ln(y)$

$div\vec{F}=20x\ln(yz)+xz+x\ln(y)$

$div\vec{F}=20x\ln(z)+yz+x\ln(y)$

$div\vec{F}=20x\ln(yz)-xz-x\ln(y)$

Correct answer:

$div\vec{F}=20x\ln(yz)+xz+x\ln(y)$

Explanation:

All we need to do is calculate the partial derivatives and add them together.

$div\vec{F}=\frac{\partial}{\partial x}(10x^2\ln(yz))+\frac{\partial}{\partial y}(xyz)+\frac{\partial}{\partial z}(xz\ln(y))$

$div\vec{F}=20x\ln(yz)+xz+x\ln(y)$

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Example Question #11 : Line Integrals

Given the vector field

$\vec{F}=yz\vec{i}+xz\vec{j}+xy\vec{k}$

find the divergence of the vector field:

$div \vec{F}$ .

Possible Answers:

$div \vec{F}=x+y+z$

$div \vec{F}=x+z$

$div \vec{F}=x+y$

$div \vec{F}=0$

$div \vec{F}=xy+yz+xz$

Correct answer:

$div \vec{F}=0$

Explanation:

Given a vector field

$\vec{F}=P\vec{i}+Q\vec{j}+R\vec{k}$

we find its divergence by taking the dot product with the gradient operator:

$div\vec{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$

We know that P=yz, Q=xz, R=xy , so we have

$div\vec{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}=0+0+0=0$

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Example Question #1 : Divergence

Suppose that $\underset{F}{\rightarrow}$ $=2x^2yz^2+y^3e^{2x}+\arctan z$ . Calculate the divergence.

Possible Answers:

$4x+3y^2+\frac{1}{1+z^2}$

$4xyz^2+2y^3e^{2x} + 2x^2z^2+3y^2e^{2x}+4x^2yz+\frac{1}{1+z^2}$

$4xyz+2x^2y^2z^2 +4x^2yz+6xy^2z+\frac{1}{1+z^2}$

$4xz^2+2y^2e^{2x} + 2x^2z+3y^2e^{2x}+4x^2yz+\frac{1}{1+z^2}$

Correct answer:

$4xyz^2+2y^3e^{2x} + 2x^2z^2+3y^2e^{2x}+4x^2yz+\frac{1}{1+z^2}$

Explanation:

We know,

$div F(x,y,z)=\frac{\partial{F}}{\partial{x}}+\frac{\partial{F}}{\partial{y}}+\frac{\partial{F}}{\partial{z}}$

Use this to obtain the correct answer

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Example Question #2 : Divergence

Given that

$\vec{F}=<2x,y,2z>$

calculate $div(\vec{F})$

Possible Answers:

$x^2+\frac{1}{2}y^2+z^2$

Correct answer:

Explanation:

$div(\vec{F})=\frac{\partial{F}}{\partial{x}}+\frac{\partial{F}}{\partial{y}}+\frac{\partial{F}}{\partial{z}}$

using this formula we have

$div(\vec{F})=2+1+2=5$

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Example Question #9 : Divergence

Find $div \vec{F}$ , where F is given by the following curve:

$\left \langle 2x, 3xy, z^2\cos(z)\right \rangle$

Possible Answers:

$2+3x+2z\cos(z)-z^2\sin(z)$

$2+3x+2z\cos(z)$

$\left \langle 2, 3x, 2z\cos(z)-z^2\sin(z)\right \rangle$

$2+3xy+2z\cos(z)-z^2\sin(z)$

Correct answer:

$2+3x+2z\cos(z)-z^2\sin(z)$

Explanation:

The divergence of a vector is given by

$div \vec{F}=\bigtriangledown \cdot \vec{F}$

where $\bigtriangledown = \left \langle f_x, f_y, f_z\right \rangle$

So, we take the partial derivative of each component of our vector with respect to x, y, and z respectively and add them together:

f_x(2x)=2

f_y(3xy)=3x

$f_z(z^2\cos(z))=2z\cos(z)-z^2\sin(z)$

The derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$

$\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$

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Example Question #3 : Divergence

Find $div \vec{F}$ where F is given by

$\left \langle 12y\cos(z), y^2e^x, z^2\right \rangle$

Possible Answers:

2ye^x+z^2

$12\cos(z)+2ye^x+2z$

2ye^x+2z

Correct answer:

2ye^x+2z

Explanation:

The divergence of a curve is given by

$div \vec{F}=\bigtriangledown \cdot \vec{F}$

where $\bigtriangledown = \left \langle f_x, f_y, f_z\right \rangle$

Taking the dot product of the gradient and the curve, we end up summing the respective partial derivatives (for example, the x coordinate's partial derivative with respect to x is found).

The partial derivatives are:

f_x=0

f_y=2ye^x

f_z=2z

The following rules were used to find the derivatives:

$\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

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Example Question #12 : Line Integrals

Compute the divergence of the vector $\left \langle xy^2,y^3z,z^3y^2\right \rangle$ .

Possible Answers:

xy+3yz+3xy^2

y^2z+xyz

y^2+3y^2z+3y^2z^2

y^2+4xy+z^3

Correct answer:

y^2+3y^2z+3y^2z^2

Explanation:

To find the divergence of the vector

$v=\left \langle a,b,c\right \rangle$ ,

we use the formula

$div(v)=\frac{\partial a}{\partial x}+\frac{\partial b}{\partial y}+\frac{\partial c}{\partial z}$ .

Computing each partial derivative, we get

$\frac{\partial }{\partial x}(xy^2)=y^2, \frac{\partial }{\partial y}(y^3z)=3y^2z,\frac{\partial }{\partial z}(y^2z^3)=3y^2z^2$ .

Adding them up gives us the correct answer.

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Example Question #13 : Line Integrals

Compute the divergence of the vector $\left \langle x^2y,xy^2z,x^4z^3\right \rangle$ .

Possible Answers:

2xyz+3x^2y+4x^3z^3

2xy+2xyz+3x^4z^2

3x^2y+4xyz+yz^3

2xy+2xyz+4x^4z^3

Correct answer:

2xy+2xyz+3x^4z^2

Explanation:

To find the divergence of the vector $v=\left \langle a,b,c\right \rangle$ , we use the formula

$div(v)=\frac{\partial a}{\partial x}+\frac{\partial b}{\partial y}+\frac{\partial c}{\partial z}$ .

Computing each partial derivative, we get

$\frac{\partial }{\partial x}(x^2y)=2xy, \frac{\partial }{\partial y}(xy^2z)=2xyz,\frac{\partial }{\partial z}(x^4z^3)=3x^4z^2$ .

Adding them up gives us the correct answer.

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Example Question #14 : Line Integrals

Compute the divergence of the vector.

$\left \langle yz,x^6y,z^5\right \rangle$

Possible Answers:

y+x^5+4z^3

y+5x^4

x^7+4z^4

x^6+5z^4

Correct answer:

x^6+5z^4

Explanation:

To find the divergence of the vector $v=\left \langle a,b,c\right \rangle$ , we use the formula

$div(v)=\frac{\partial a}{\partial x}+\frac{\partial b}{\partial y}+\frac{\partial c}{\partial z}$ .

Computing each partial derivative, we get

$\frac{\partial }{\partial x}(yz)=0, \frac{\partial }{\partial y}(x^6y)=x^6,\frac{\partial }{\partial z}(z^5)=5z^4$ .

Adding them up gives us the correct answer.

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Example Question #15 : Line Integrals

Find the divergence of the vector $\left \langle 2x^3yz,3y^3,z^8\right \rangle$

Possible Answers:

$\left \langle 6x^2yz,9y^2,8z^7\right \rangle$

5x^2yz+y^4+8z^7

6x^2yz+9y^2+8z^7

x^6yz+4y^3+z^8

Correct answer:

6x^2yz+9y^2+8z^7

Explanation:

The formula for the divergence of a vector $a=\left \langle A,B,C\right \rangle$ is $div(a)=\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}$ . Using the vector from the problem statement, we get $\frac{\partial }{\partial x}(2x^3yz)=6x^2yz, \frac{\partial }{\partial y}(3y^3)=9y^2,\frac{\partial }{\partial z}(z^8)=8z^7$ . Adding them up gets us the correct answer.

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Calculus 3 : Line Integrals

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1 : Divergence

Example Question #11 : Line Integrals

Example Question #1 : Divergence

Example Question #2 : Divergence

Example Question #9 : Divergence

Example Question #3 : Divergence

Example Question #12 : Line Integrals

Example Question #13 : Line Integrals

Example Question #14 : Line Integrals

Example Question #15 : Line Integrals

All Calculus 3 Resources

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