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Calculus 3 : Divergence

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Example Questions

Example Question #11 : 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^2+3y^2z+3y^2z^2

y^2+4xy+z^3

y^2z+xyz

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 #12 : 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 #13 : Line Integrals

Compute the divergence of the vector.

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

Possible Answers:

y+x^5+4z^3

x^6+5z^4

x^7+4z^4

y+5x^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 #14 : Line Integrals

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

Possible Answers:

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

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

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

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

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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Example Question #21 : Line Integrals

Find the divergence of the vector $\left \langle x^2y^2z^2,3y^5,17z\right \rangle$

Possible Answers:

x^2y^2z^2+15y^5-17z

xy^2z^2+y^5+z

3xyz+3y^5-17z

2xy^2z^2+15y^4+17

Correct answer:

2xy^2z^2+15y^4+17

Explanation:

To find the divergence a vector $a=\left \langle A,B,C\right \rangle$ , you use the following definition: $div(a)=\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}$ . Applying this to the vector from the problem statement, we get $\frac{\partial A}{\partial x}=2xy^2z^2, \frac{\partial B}{\partial y}=15y^4,\frac{\partial C}{\partial z}=17$ . Adding all of these up, according to the definition, will produce the correct answer.

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Example Question #22 : Line Integrals

Find the divergence of the following vector: $\left \langle yz^3,xy^5z,3xz\right \rangle$

Possible Answers:

$\left \langle 0,5xy^4z,3x\right \rangle$

$\left \langle 3yz^2,5xy^3z,3z\right \rangle$

$\left \langle 0,5xy^4z,3\right \rangle$

$\left \langle 3yz^2,y^5z,3x\right \rangle$

Correct answer:

$\left \langle 0,5xy^4z,3x\right \rangle$

Explanation:

To find the divergence a vector $a=\left \langle A,B,C\right \rangle$ , you use the following definition: $div(a)=\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}$ . Applying this to the vector from the problem statement, we get $\frac{\partial A}{\partial x}=0, \frac{\partial B}{\partial y}=5xy^4z,\frac{\partial C}{\partial z}=3x$ . Adding all of these up, according to the definition, will produce the correct answer.

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Example Question #1671 : Calculus 3

Find the divergence of the vector $\left \langle \cos(x),-2\sin(y),x^2y^2z^2\right \rangle$

Possible Answers:

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

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

$\left \langle \sin(x),\cos(y),x^2y^2\right \rangle$

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

Correct answer:

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

Explanation:

To find the divergence a vector $a=\left \langle A,B,C\right \rangle$ , you use the following definition: $div(a)=\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}$ . Applying this to the vector from the problem statement, we get $\frac{\partial A}{\partial x}=-\sin(x), \frac{\partial B}{\partial y}=-2\cos(x),\frac{\partial C}{\partial z}=2x^2y^2z$ . Adding all of these up, according to the definition, will produce the correct answer.

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Example Question #1672 : Calculus 3

Find the divergence of the vector $\left \langle 2x,6x^2y^3,z^4\right \rangle$

Possible Answers:

2+18xy^2+4z^3

2+6xy^3+z^5

2+18x^2y^2+4z^3

18x^2y^3+4z^3

Correct answer:

2+18x^2y^2+4z^3

Explanation:

To find the divergence of a vector $v=\left \langle A,B,C\right \rangle$ , we apply the following definition: $div(v)=\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}$ . Applying the definition to the vector from the problem statement, we get $\frac{\partial }{\partial x}(2x)+\frac{\partial }{\partial y}(6x^2y^3)+\frac{\partial }{\partial z}(z^4)=2+18x^2y^2+4z^3$

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Example Question #1673 : Calculus 3

Find the divergence of the vector $\left \langle xz^4,6x^5y,x^4yz^2\right \rangle$

Possible Answers:

z^4+6x^5+x^4y

z^4+30x^6+8x^3yz

z^4+6x^5+2x^4yz

z^4+5x^4+2x^3yz^2

Correct answer:

z^4+6x^5+2x^4yz

Explanation:

To find the divergence of a vector $v=\left \langle A,B,C\right \rangle$ , we apply the following definition: $div(v)=\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}$ . Applying the definition to the vector from the problem statement, we get $\frac{\partial }{\partial x}(xz^4)+\frac{\partial }{\partial y}(6x^5y)+\frac{\partial }{\partial z}(x^4yz^2)=z^4+6x^5+2x^4yz$

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Example Question #1674 : Calculus 3

Find the divergence of the vector $\left \langle 12x^2y,4y^2z,xy^5\right \rangle$

Possible Answers:

12x+8y

24xy+8yz

3xy+2yz+z

12x+4xyz

Correct answer:

24xy+8yz

Explanation:

To find the divergence of a vector $v=\left \langle A,B,C\right \rangle$ , we apply the following definition: $div(v)=\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}$ . Applying the definition to the vector from the problem statement, we get $\frac{\partial }{\partial x}(12x^2y)+\frac{\partial }{\partial y}(4y^2z)+\frac{\partial }{\partial z}(xy^5)=24xy+8yz$

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Calculus 3 : Divergence

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #11 : Line Integrals

Example Question #12 : Line Integrals

Example Question #13 : Line Integrals

Example Question #14 : Line Integrals

Example Question #21 : Line Integrals

Example Question #22 : Line Integrals

Example Question #1671 : Calculus 3

Example Question #1672 : Calculus 3

Example Question #1673 : Calculus 3

Example Question #1674 : Calculus 3

All Calculus 3 Resources

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