Objectives
- Examine the geometric concepts related to the gradient.
- Explore how the gradient can be used to move towards a maximum or minimum.
Section 6.8 Gradients
This lesson defines the gradient of a multivariable function as the vector with components given by the partial derivatives. Further, the gradient is the first example of a vector field, a multivariable function with vector outputs. The two main geometric properties of the gradient are demonstrated over several examples including surfaces without symmetries. In these examples, contour plots are plotted to show the relationship between level curves the direction of the gradient.
Subsection 6.8.1 Exploring the Geometry Connected to the Gradient
Activity 6.8.1.
Consider the function
\begin{equation*}
f(x,y) =\frac{4x}{1+x^2+y^2}
\end{equation*}
and starting point \((-1,2)\) shown below:
(a)
Find the gradient at the function \(f\text{.}\)
(b)
Find the value of the gradient at the starting point.
(c)
From the starting point move one unit in the direction of the gradient. What point are you at? Add the point to the graph as Point 1.
Hint.
Remember you can scale a vector to unit length by dividing by the magnitude. Adding this vector to the point will move you to the desired location.
(d)
Repeat the previous two steps using Point 1 as your starting point. What point have you moved to? Add the point to the graph as Point 2.
(e)
Repeat the previous two steps using Point 2 as your starting point. What point have you moved to? Add the point to the graph as Point 3.
(f)
Describe where you are moving towards. Explain why this is the case using properties of the gradient.
Exercises 6.8.2 Exercises
1.
Evaluate the cell below to create a contour plot. At each indicated point on the following contour plot draw arrows in the direction of the gradient.
2.
You are standing at a point where the temperature gradient vector is pointing in the northeast (NE) direction. In which direction(s) should you walk to avoid a change in temperature.
3.
The figure below shows the level curves of a function \(f\) an a path \(\mathbf{r}\text{,}\) traversed from left to right. State whether the derivative \(\frac{d}{dt}f(\mathbf{r}(t)) \) is positive, negative, or zero at the indicated point.
