# Vector Differential Calculus Questions and Answers – Using Properties of Divergence and Curl

This set of Engineering Mathematics Problems focuses on “Using Properties of Divergence and Curl”.

1. Find the divergence of this given vector $$\vec{F}=x^3 y\vec{i}+3xy^2 z\vec{j}+3zx\vec{k}$$.
a) 3x2 y+6xyz+x
b) 2x2 y+6xyz+3x
c) 3x2 y+3xyz+3x
d) 3x2 y+6xyz+3x

Explanation: We know that divergence of a vector is given by
$$\bigtriangledown.\vec{F}=\frac{\partial(x^3 y)}{\partial x}+\frac{\partial(3xy^2 z)}{\partial y}+\frac{\partial(3xz)}{\partial z}$$
$$\bigtriangledown.\vec{F}=3x^2 y+6xyz+3x$$.

2. Find the divergence of this given vector $$\vec{r}=12x^6 y^6 \vec{i}+3x^3 y^3 z\vec{j}+3x^2 yz^2 \vec{k}$$.
a) $$12x^5 y^6+2x^3 yz+6x^2 yz$$
b) $$72x^5 y^6+2x^3 yz+3x^2 yz$$
c) $$72x^5 y^6+2x^3 yz+6x^2 yz$$
d) $$6x^5 y^6+2x^3 yz+6x^2 yz$$

Explanation: We know that divergence of a vector is given by
$$\bigtriangledown.\vec{r}=\frac{\partial(12x^6 y^6)}{\partial x}+\frac{\partial(3x^3 y^3 z)}{\partial y}+\frac{\partial(3x^2 yz^2)}{\partial z}$$
$$\bigtriangledown.\vec{r}=12×6x^5 y^6+x^3×2y×z+3x^2×y×2z$$
$$\bigtriangledown.\vec{r}=72x^5 y^6+2x^3 yz+6x^2 yz$$.

3. Find the curl for $$\vec{r}=x^2 yz\vec{i}+(3x+2y)z\vec{j}+21z^2 x\vec{k}$$.
a) $$\vec{i}(3x+2y)-\vec{j}(11z^2-x^2 y)+\vec{k}(3z-x^2 z)$$
b) $$\vec{i}(x+2y)-\vec{j}(21z^2-x^2 y)+\vec{k}(3z-x^2 z)$$
c) $$-\vec{i}(3x+2y)-\vec{j}(21z^2-x^2 y)+\vec{k}(3z-x^2 z)$$
d) $$\vec{i}(3x+2y)-\vec{j}(21z^2-x^2 y)+\vec{k}(3z-x^2 z)$$

Explanation: We know that the curl for any vector quantity is given by
$$\bigtriangledown.\vec{r}=\begin{bmatrix}\vec{i}&\vec{j}&\vec{k}\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\x^2 yz&(3x+2y)z&21z^2 x\end{bmatrix}$$
$$\bigtriangledown.\vec{r}=\vec{i}\left (\frac{\partial(21z^2 x)}{\partial y}-\frac{\partial((3x+2y)z)}{\partial z}\right)-\vec{j}\left (\frac{\partial(21z^2 x)}{\partial x}-\frac{\partial(x^2 yz)}{\partial z}\right)$$
$$+\vec{k}\left (\frac{\partial((3x+2y)z)}{\partial x}-\frac{\partial(x^2 yz)}{\partial y}\right)$$
$$\bigtriangledown.\vec{r}=-\vec{i}(3x+2y)-\vec{j}(21z^2-x^2 y)+\vec{k}(3z-x^2 z)$$.

4. Find the curl for $$(\vec{r})=y^2 z^3 \vec{i}+x^2 z^2 \vec{j}+(x-2y)\vec{k}$$.
a) $$-2\vec{i}(1+x^2 z)-\vec{j}(1-3y^2 z^2)+\vec{2k}(xz^2-yz^3)$$
b) $$-2\vec{i}(1+x^2 z)-\vec{j}(1-3y^2 z^2)+\vec{k}(xz^2-yz^3)$$
c) $$-2\vec{i}(1+x^2 z)-\vec{j}(1-32z^2)+\vec{2k}(xz^2-yz^3)$$
d) $$\vec{i}(1+x^2 z)-\vec{j}(1-3y^2 z^2)+\vec{2k}(xz^2-yz^3)$$

Explanation: We know that the curl for any vector quantity is given by
$$\bigtriangledown.\vec{r}=\begin{bmatrix}\vec{i}&\vec{j}&\vec{k}\\\partial/\partial x&\partial/\partial y&\partial/\partial z\\y^2 z^3&x^2 z^2&(x-2y)\end{bmatrix}$$
$$\bigtriangledown.\vec{r}=\vec{i}\left (\frac{\partial(x-2y)}{\partial y}-\frac{\partial(x^2 z^2)}{/\partial z}\right )-\vec{j}\left (\frac{\partial(x-2y)}{\partial x}-\frac{\partial(y^2 z^3)}{\partial z}\right )+\vec{k}\left(\frac{\partial(x^2 z^2)}{\partial x}-\frac{\partial(y^2 z^3)}{\partial y}\right )$$ $$\bigtriangledown.\vec{r}=-2\vec{i}(1+x^2 z)-\vec{j}(1-3y^2 z^2)+\vec{2k}(xz^2-yz^3)$$.

5. What is the divergence and curl of the vector $$\vec{F}=x^2 y\vec{i}+(3x+y) \vec{j}+y^3 z\vec{k}$$.
a) $$y^3+2xy+1,\vec{i}(3y^2 z)+\vec{j}(3-x^2)$$
b) $$y^3+2xy+1,\vec{i}(3y^2 z)+\vec{k}(3-x^2)$$
c) $$3y^3+2xy+1,\vec{i}(3y^2 z)+\vec{k}(3-x^2)$$
d) $$y^3+xy+1,\vec{i}(3y^2 z)+\vec{k}(3-x^2)$$

Explanation: We know that divergence of a vector is given by
$$\bigtriangledown.\vec{F}=\frac{\partial(x^2 y)}{\partial x}+\frac{\partial(3x+y)}{\partial y}+\frac{\partial(y^3 z)}{\partial z}$$
$$\bigtriangledown.\vec{F}=y^3+2xy+1$$
We know that the curl for any vector quantity is given by
$$\bigtriangledown.\vec{r}=\begin{bmatrix}\vec{i}&\vec{j}&\vec{k}\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\x^2 y&3x+y&(y^3 z)\end{bmatrix}$$
$$\bigtriangledown.\vec{r}=\vec{i}\left (\frac{\partial(y^3 z)}{\partial y}-\frac{\partial(3x+y)}{\partial z}\right )-\vec{j}\left (\frac{\partial(y^3 z)}{\partial x}-\frac{\partial(x^2 y)}{\partial z}\right )+\vec{k}\left (\frac{\partial(3x+y)}{\partial x}-\frac{\partial(x^2 y)}{\partial y}\right )$$
$$\bigtriangledown.\vec{r}=\vec{i}(3y^2 z)+\vec{k}(3-x^2)$$.
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