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Calculus 3 : Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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 #51 : Divergence

Find the divergence of the vector field:

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

Possible Answers:

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

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

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

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

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

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 #52 : Divergence

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

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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

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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 , $f_z=-xy\sin(xyz)$

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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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Example Question #65 : Divergence

Find the divergence of the vector $\left \langle 2xy^4,y^5,\tan(z)\right \rangle$

Possible Answers:

$7y^3+\cos^2(z)$

$7y^4+\sec^2(z)$

$y^4-\sec^2(z)$

$7y^4+\sec^2(x)$

Correct answer:

$7y^4+\sec^2(z)$

Explanation:

To find the divergence of a vector $v=\left \langle P,Q,R\right \rangle$ , we use the formula:

$div(v)=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$

Applying to the vector from the problem statement, we get

$\frac{\partial }{\partial x}(2xy^4)+\frac{\partial }{\partial y}(y^5)+\frac{\partial }{\partial z}(\tan(z))=7y^4+\sec^2(z)$

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Example Question #66 : Divergence

Find the divergence of the vector $\left \langle \sin(x),5yz^3,\cos(z)\right \rangle$

Possible Answers:

$\cos(x)+4z^3-\sin(z)$

$-\cos(x)+5z^3-\sin(z)$

$\cos(x)+5z^3-\sin(z)$

$\cos(x)+5z^3+\sin(z)$

Correct answer:

$\cos(x)+5z^3-\sin(z)$

Explanation:

To find the divergence of a vector $v=\left \langle P,Q,R\right \rangle$ , we use the formula:

$div(v)=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$

Applying to the vector from the problem statement, we get

$\frac{\partial }{\partial x}(\sin(x))+\frac{\partial }{\partial y}(5yz^3)+\frac{\partial }{\partial z}(\cos(z))=\cos(x)+5y^3-\sin(z)$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #63 : Line Integrals

Example Question #51 : Divergence

Example Question #65 : Line Integrals

Example Question #52 : Divergence

Example Question #61 : Divergence

Example Question #62 : Divergence

Example Question #63 : Divergence

Example Question #64 : Divergence

Example Question #65 : Divergence

Example Question #66 : Divergence

All Calculus 3 Resources

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