Gradient Vector

The Chain Rule for Paths

Given a function , notice how we can view this as a function of only, since and are directly dependent on .

Now, recall from the chain rule for single variable functions. We can generalize this for multivariable functions as well:

Definition

Given :

(see Partial Derivatives)

Example

Suppose that satisfies the partial differential equation:

By setting and , there is a constant such that

Determine the value of .

solution
Using the chain rule,

Therefore

The Gradient Vector

Definition

We define the gradient vector as:

etc.

Definition

We define the gradient vector of as

read "grad " or "del " or "nabla ".

We can re-write the chain rule with the gradient vector:

Equation

The chain rule for paths can also be written:

We can also re-write the equation of a tangent plane:

With the Gradient Vector

(See Gradient Vector)

We can develop a short hand to write the tangent plane at the point , with .

(see Dot Product)

The components of the gradient vector give us the slope of tangent plane in the x, y directions respectively.

Equation

The linear approximation of the point knowing is:

Interpretation of the Gradient Vector Itself

How do we determine the maximum and minimum steepness of a certain surface? Recall the dot product is really the magnitude of a projection:

Max steepness:
Max negative steepness:
Zero steepness:

This means that when is the same direction as , we have the max steepness, which is .

Visualization

In the direction of , slope is max positive, .
In the direction of , the slope is min
Along contour lines/level curves:

Fact

The gradient vector is always perpendicular to the contour lines

Example

Height of a (very high) mountain is modelled by:

At the point , what direction is the stream flowing? I.e direction of max negative .

solution

Gradient:

The stream is flowing in the opposite direction (our vector currently points towards the centre going up):

Note that the units are of are actually , aka a "grade" (engineers) (m down per m moved in the x or y direction). You can see from the derivative we took, , we have change in height with respect to , not time.

Gradient vector can also be used to visualize a surface. Each vector represents the slope of the surface at that point

Example

Give the "vector representation" of the following function:

solution
This is the contour plot:

Taking the partial derivatives to find our gradient vector: $$ \vec{\nabla}f = (-2x, 2y) $$

Now, if we chose a bunch of points and plot this vector, we get something like this: