Find the divergence of the vector field. $\overrightarrow { \mathbf { F } } ( x , y , z ) = 10 x ^ { 6 } \hat { \mathbf { i } } - x y ^ { 5 } \hat { \mathbf { j } }$
A) $\operatorname { div } \overrightarrow { \mathrm { F } } ( x , y , z ) = 60 x ^ { 7 } - 6 x y ^ { 6 }$
B) $\quad \operatorname { div } \overrightarrow { \mathrm { F } } ( x , y , z ) = 60 x ^ { 5 } - 5 x y ^ { 4 }$
C) $\operatorname { div } \overrightarrow { \mathrm { F } } ( x , y , z ) = 10 x ^ { 6 } \hat { \mathbf { i } } - 6 x y ^ { 5 } \hat { \mathbf { j } }$
D)$\operatorname { div } \overrightarrow { \mathrm { F } } ( x , y , z ) = \frac { 10 } { 7 } x ^ { 7 } - \frac { 1 } { 6 } x y ^ { 6 }$
E)$\operatorname { div } \overrightarrow { \mathrm { F } } ( x , y , z ) = \frac { 10 } { 7 } x ^ { 7 } \hat { \mathbf { i } } - \frac { 1 } { 5 } x y ^ { 4 } \hat { \mathbf { j } }$

Question

Find the divergence of the vector field. $\overrightarrow { \mathbf { F } } ( x , y , z ) = 10 x ^ { 6 } \hat { \mathbf { i } } - x y ^ { 5 } \hat { \mathbf { j } }$
A) $\operatorname { div } \overrightarrow { \mathrm { F } } ( x , y , z ) = 60 x ^ { 7 } - 6 x y ^ { 6 }$
B) $\quad \operatorname { div } \overrightarrow { \mathrm { F } } ( x , y , z ) = 60 x ^ { 5 } - 5 x y ^ { 4 }$ 
C) $\operatorname { div } \overrightarrow { \mathrm { F } } ( x , y , z ) = 10 x ^ { 6 } \hat { \mathbf { i } } - 6 x y ^ { 5 } \hat { \mathbf { j } }$
D)$\operatorname { div } \overrightarrow { \mathrm { F } } ( x , y , z ) = \frac { 10 } { 7 } x ^ { 7 } - \frac { 1 } { 6 } x y ^ { 6 }$
E)$\operatorname { div } \overrightarrow { \mathrm { F } } ( x , y , z ) = \frac { 10 } { 7 } x ^ { 7 } \hat { \mathbf { i } } - \frac { 1 } { 5 } x y ^ { 4 } \hat { \mathbf { j } }$

Quizplus · Accepted Answer

The divergent of a vector field $\overrightarrow{\mathrm{F}} = f_1(x,y,z)\hat{\mathrm{i}} + f_2(x,y,z)\hat{\mathrm{j}} + f_3(x,y,z)\hat{\mathrm{k}}$ is defined as
$$\operatorname{div}\overrightarrow{\mathrm{F}} = \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} + \frac{\partial f_3}{\partial z}$$
In our case, $\overrightarrow{\mathrm{F}}(x,y,z) = 10x^6\hat{\mathrm{i}} - xy^5\hat{\mathrm{j}} + 0\hat{\mathrm{k}}$, so
$$\frac{\partial \overrightarrow{\mathrm{F}}}{\partial x} = 10^6\hat{\mathrm{i}} - y^5\hat{\mathrm{j}} + 0\hat{\mathrm{k}}, \quad \frac{\partial \overrightarrow{\mathrm{F}}}{\partial y} = 0\hat{\mathrm{i}} - 5xy^4\hat{\mathrm{j}} + 0\hat{\mathrm{k}}, \quad \frac{\partial \overrightarrow{\mathrm{F}}}{\partial z} = 0\hat{\mathrm{i}} + 0\hat{\mathrm{j}} + 0\hat{\mathrm{k}}$$
Therefore, the divergence of $\overrightarrow{\mathrm{F}}$ is
$$\operatorname{div}\overrightarrow{\mathrm{F}} = \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} + \frac{\partial f_3}{\partial z} = 60x^5 - 5xy^4$$
So the answer is B.

Find the Divergence of the Vector Field F→(x,y,z)=10x6i^−xy5j^\overrightarrow { \mathbf { F } } ( x , y , z ) = 10 x ^ { 6 } \hat { \mathbf { i } } - x y ^ { 5 } \hat { \mathbf { j } }F(x,y,z)=10x6i^−xy5j^​

Find the Divergence of the Vector Field $\overrightarrow { \mathbf { F } } ( x , y , z ) = 10 x ^ { 6 } \hat { \mathbf { i } } - x y ^ { 5 } \hat { \mathbf { j } }$