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Calculus 3 : Line Integrals

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Example Questions

Example Question #61 : Line Integrals

Find the divergence of the vector field:

$\vec{F}=\left \langle x^5e^y, e^{xy}, z\right \rangle$

Possible Answers:

$5x^4+xe^{xy}+1$

$5x^4e^y+ye^{xy}+1$

$5x^4e^y+xe^{xy}+1$

$5x^4e^y+xe^{xy}+z$

Correct answer:

$5x^4e^y+xe^{xy}+1$

Explanation:

The divergence of the vector field

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

where $\bigtriangledown = \left \langle \frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}\right \rangle$

In taking the dot product, we get the sum of the respective partial derivatives of the vector field.

The partial derivatives are

f_x=5x^4e^y , $f_y=xe^{xy}$ , f_z=1

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Example Question #62 : Line Integrals

Find the divergence of the following vector field:

$\vec{F}=\left \langle e^{\tan(y)}, x^3y, x^2\right \rangle$

Possible Answers:

x^3

$\sec^2(y)e^{\tan(y)}+3x^2y+2x$

3x^2y

Correct answer:

x^3

Explanation:

The divergence of a vector field is given by

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

where $\bigtriangledown = \left \langle \frac{\partial }{\partial x}, \frac{\partial }{\partial y} , \frac{\partial }{\partial z}\right \rangle$

When we take the dot product of the gradient with the vector field, we get the sum of the respective partial derivatives of the vector field.

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

f_x=0 , f_y=x^3 , f_z=0

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Example Question #63 : Line Integrals

Determine the divergence of the vector field:

$\vec{F}=\left \langle x^4y^2, e^{\cos(y)}, x+y\right \rangle$

Possible Answers:

$4x^3y^2-\sin(y)e^{\cos(y)}$

$4x^3y^2-\sin(y)e^{\cos(y)}+x+y$

$4x^3y^2+\sin(y)e^{\cos(y)}$

$4x^3-\sin(y)e^{\cos(y)}$

Correct answer:

$4x^3y^2-\sin(y)e^{\cos(y)}$

Explanation:

The divergence of a vector field is given by

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

where $\bigtriangledown = \left \langle \frac{\partial }{\partial x} , \frac{\partial }{\partial y}, \frac{\partial }{\partial z}\right \rangle$

In taking the dot product, we get the sum of the respective partial derivatives of the vector field.

The partial derivatives are

f_x=4x^3y^2 , $f_y=-\sin(y)e^{\cos(y)}$ , f_z=0

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Example Question #61 : Line Integrals

Find the divergence of the vector field:

$\vec{F}=\left \langle 5\cos(y), 6x, z^2e^{3\tan(z)}\right \rangle$

Possible Answers:

$2ze^{3\tan(z)}+6z\sec^2e^{3\tan(z)}$

$2ze^{3\tan(z)}+6z^2\sec^2e^{3\tan(z)}$

$2ze^{3\tan(z)}+3z^2\sec^2e^{3\tan(z)}$

$6ze^{3\tan(z)}+6z^2\sec^2e^{3\tan(z)}$

$2ze^{3\tan(z)}+6z^2\sec(z)e^{3\tan(z)}$

Correct answer:

$2ze^{3\tan(z)}+6z^2\sec^2e^{3\tan(z)}$

Explanation:

The divergence of a vector field is given by

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

where $\bigtriangledown = \left \langle \frac{\partial }{\partial x} , \frac{\partial }{\partial y}, \frac{\partial }{\partial z}\right \rangle$

In taking the dot product, we get the sum of the respective partial derivatives of the vector field.

The partial derivatives are

f_x=0 , f_y=0 , $f_z=2ze^{3\tan(z)}+6z^2\sec^2e^{3\tan(z)}$

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Example Question #65 : Line Integrals

Find the divergence of the following vector field:

$\vec{F}=\left \langle xy\sec(x), y^2, \cos(y)\right \rangle$

Possible Answers:

$y\sec(x)(1+x\tan(x))+2y$

$\sec(x)(1+x\tan(x))+3y$

$y\sec(x)(1+x\sec(x)\tan(x))+2y$

$\sec(x)(1+x\tan(x))+2y$

Correct answer:

$y\sec(x)(1+x\tan(x))+2y$

Explanation:

The divergence of a vector field is given by $\bigtriangledown \cdot \vec{F}$ , where $\bigtriangledown=\left \langle \frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}\right \rangle$

When we take the dot product of the gradient with the vector field, we are left with the sum of the respective partial derivative of the vector field.

The partial derivatives are

$f_x=y\sec(x)(1+x\tan(x))$ , f_y=2y , f_z=0

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Example Question #62 : Line Integrals

Find the divergence of the vector field:

$\vec{F}=\left \langle e^{\cos(xy)}, z^2, z^3\right \rangle$

Possible Answers:

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

$-y\sin(xy)e^{\cos(xy)}+4z^2$

$-y\sin(xy)e^{\cos(xy)}+3z^2$

$y\sin(xy)e^{\cos(xy)}+3z^2$

Correct answer:

$-y\sin(xy)e^{\cos(xy)}+3z^2$

Explanation:

The divergence of a vector field is given by

$div \vec{F}=\bigtriangledown \cdot \vec{F}$ , where $\bigtriangledown = \left \langle \frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z} \right \rangle$

When we take the dot product, we get the sum of the respective partial derivatives of the vector field. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

$f_x=-y\sin(xy)e^{\cos(xy)}$ , f_y=0 , f_z=3z^2

Report an Error

Example Question #61 : Divergence

Find the divergence of the vector field:

$\vec{F}=\left \langle \tan(x^2z), y^2, xy\right \rangle$

Possible Answers:

$-2xz\sec^2(x^2z)+2y$

$2xz\sec^2(x^2z)+y^2$

$2xz\sec^2(x^2z)+2y$

$x^2z\sec^2(x^2z)+2y$

Correct answer:

$2xz\sec^2(x^2z)+2y$

Explanation:

The divergence of a vector field is given by

$div \vec{F}=\bigtriangledown \cdot \vec{F}$ , where $\bigtriangledown = \left \langle \frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z} \right \rangle$

When we take the dot product of the gradient and the vector field, we get the sum of the respective partial derivatives. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

$f_x=2xs\sec^2(x^2z)$ , f_y=2y , f_z=0

Report an Error

Example Question #62 : Divergence

Find the divergence of the vector field:

$\vec{F}=\left \langle xe^{x^2}, yz^3, z^2\right \rangle$

Possible Answers:

$e^{x^2}+2x^2e^{x^2}+z^3+2z$

$e^{x^2}+2xe^{x^2}+z^3+2z$

$e^{x^2}+z^3+2z$

$2x^2e^{x^2}+z^3+2z$

Correct answer:

$e^{x^2}+2x^2e^{x^2}+z^3+2z$

Explanation:

The divergence of a vector field is given by

$div \vec{F}=\bigtriangledown \cdot \vec{F}$ , where $\bigtriangledown = \left \langle \frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z} \right \rangle$

The partial derivatives are

$f_x=e^{x^2}+2x^2e^{x^2}$ , f_y=z^3 , f_z=2z

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Example Question #63 : Divergence

Find the divergence of the vector field:

$\vec{F}=\left \langle x^3z^4, ye^y+y^2z, \cos(xyz)\right \rangle$

Possible Answers:

$3x^2z^4+e^y(1+y)+2yz-xy\sin(xyz)$

$3x^2z^4+e^y(1+y)-xy\sin(xyz)$

$3x^2z^4+e^y(1+y)+2yz+xy\sin(xyz)$

$3x^2z^4+ye^y(1+y)+2yz-xy\sin(xyz)$

Correct answer:

$3x^2z^4+e^y(1+y)+2yz-xy\sin(xyz)$

Explanation:

The divergence of a vector field is given by

$div\vec{F}=\bigtriangledown \cdot \vec{F}$ , where $\bigtriangledown = \left \langle \frac{\partial }{\partial x}, \frac{\partial }{\partial y} , \frac{\partial }{\partial z}\right \rangle$

In taking the dot product of the gradient with the vector field, we get the sum of the respective partial derivatives. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

f_x=3x^2z^4 , f_y=e^y(1+y)+2yz , $f_z=-xy\sin(xyz)$

Report an Error

Example Question #64 : Divergence

Find the divergence of the vector field:

$\vec{F}=\left \langle x^2+3z, \cos^2(yz), z\sec(x)\right \rangle$

Possible Answers:

$2x+3-2\cos(yz)\sin(yz)+\sec(x)$

$2x+3+2z\cos(yz)\sin(yz)+\sec(x)$

$5x-2z\cos(yz)\sin(yz)+\sec(x)$

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

Correct answer:

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

Explanation:

The divergence of a vector field is given by

$div \vec{F}=\bigtriangledown \cdot \vec{F}$ , where $\bigtriangledown = \left \langle \frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z} \right \rangle$ .

The partial derivatives are

f_x=2x+3 , $f_y=-2z\cos(yz)\sin(yz)$ , $f_z=\sec(x)$

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Calculus 3 : Line Integrals

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #61 : Line Integrals

Example Question #62 : Line Integrals

Example Question #63 : Line Integrals

Example Question #61 : Line Integrals

Example Question #65 : Line Integrals

Example Question #62 : Line Integrals

Example Question #61 : Divergence

Example Question #62 : Divergence

Example Question #63 : Divergence

Example Question #64 : Divergence

All Calculus 3 Resources

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