Find the gradient vector field of the function $f ( x , y ) = x y ^ { 2 }$ .
A) $\mathbf { i } + 2 \mathbf { j }$
B)$\mathbf { i } - 2 \mathbf { j }$
C) $x \mathbf { i } + y ^ { 2 } \mathbf { j }$
D)$x \mathbf { i } - y ^ { 2 } \mathbf { j }$
E)$\mathbf { i } + 2 y \mathbf { j }$
F)$\mathbf { i } - 2 y \mathbf { j }$
G) $y ^ { 2 } \mathbf { i } + 2 x y \mathbf { j }$
H)$y ^ { 2 } \mathbf { i } - 2 x y \mathbf { j }$

Question

Find the gradient vector field of the function $f ( x , y ) = x y ^ { 2 }$ .
A) $\mathbf { i } + 2 \mathbf { j }$
B)$\mathbf { i } - 2 \mathbf { j }$ 
C) $x \mathbf { i } + y ^ { 2 } \mathbf { j }$
D)$x \mathbf { i } - y ^ { 2 } \mathbf { j }$ 
E)$\mathbf { i } + 2 y \mathbf { j }$ 
F)$\mathbf { i } - 2 y \mathbf { j }$ 
G) $y ^ { 2 } \mathbf { i } + 2 x y \mathbf { j }$ 
H)$y ^ { 2 } \mathbf { i } - 2 x y \mathbf { j }$

Quizplus · Accepted Answer

The gradient vector field of a function $f(x, y)$ is given by $\nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j}$. For $f(x, y) = xy^2$, $\frac{\partial f}{\partial x} = y^2$ and $\frac{\partial f}{\partial y} = 2xy$, thus the gradient is $y^2 \mathbf{i} + 2xy \mathbf{j}$.

Find the Gradient Vector Field of the Function f(x,y)=xy2f ( x , y ) = x y ^ { 2 }f(x,y)=xy2

Find the Gradient Vector Field of the Function $f ( x , y ) = x y ^ { 2 }$