## Related questions with answers

The force in an electrostatic field given by f(x, y, z) has the direction of the gradient. Find ∇f and its value at P.

Solutions

VerifiedGiven a differentiable function $f:\mathbb{R}^2\to\mathbb{R}$, it's *gradient* $\nabla f$ is a function from $\mathbb{R}^2$ to $\mathbb{R}^2$. At a point $(x,y)$, the direction of $\nabla f(x,y)$ points to the direction of the greatest growth of $f$, and it's magnitude tells us the speed of this growth.

The gradient of a function is defined as

$\nabla=\frac{\partial}{\partial x}\textbf{i}+\frac{\partial}{\partial y}\textbf{j}= \Big(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\Big)$

Let's find partial derivates.

$\begin{align*} \frac{\partial}{\partial x} &= y\\ \frac{\partial}{\partial y} &=x \end{align*}$

Consequently,

$\boxed{ \nabla f=\Big(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\Big)=(y, x)}$

Since $p=(-4,5)$ we have:

$\color{#c34632}{ \nabla f (p)= (5,-4)}$

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