Directional derivatives and the gradient vector

Recall that the directional derivative of $f$ at $(a,b)$ in the direction of an arbitrary nonzero vector $\mathbf u =(u_1,u_2)$ is given by

\begin{eqnarray*} D_u f(a,b)&=&f_{\mathbb u}(a,b)=\big( f_x(a,b),f_y(a,b)\big)\cdot \frac{\mathbb u}{||\mathbb u||} \end{eqnarray*}

On the other hand, the gradient vector or simply gradient of $f$ is a vector with the partial derivatives as components

\begin{eqnarray*} \text{grad}\, f&=& \nabla f =\big( f_x,f_y\big)=f_x\,\mathbf i+f_y\,\mathbf j. \end{eqnarray*}

Simulation

This simulation shows the geometric interpretation of the directional derivative of $f$ in the direction of a unit vector $\mathbf u$ and the gradient vector of $f(x,y)$ at the point $\text{P}\in \mathbb R^2$.

Things to try:

Change the function f(x,y). Example: f(x,y)= y^2-(x-1)^2
Drag the point P or type specific values on the boxes.
Activate box Dir. derivative to show the directional derivative. Change the angle $\theta$ by dragging the slider or type an specific value on the box.
Use right click and drag the mouse to rotate the 3D view or click on View button.

Note: The blue arrow represents the gradient vector at point P.

Sorry, the simulation is not supported for small screens.

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