Vector field I: gradient of a scalar function

A vector field can be derived from a scalar function by computing its gradient. In this example, we define a scalar function:

The gradient of this function gives us a vector field where each vector points in the direction of steepest ascent. The gradient is:

Let's visualize this vector field and see how changing the parameter a affects it.

First, we define the parameter a:

• Define parameter a: a = \scalar 1

Now we need a point to evaluate the function and its gradient. Let's start with a point p:

• Create point p: p = \point 0 0

Setting up the inputs and output

To compute , we need to extract the x and y coordinates:

• Get x coordinate: px = \x-coord p
• Get y coordinate: py = \y-coord p

Now we calculate the squared terms:

• Calculate : px-squared = \mul px px
• Calculate : py-squared = \mul py py

Multiply a by :

• Calculate : a-px-squared = \mul a px-squared

Finally, compute :

• Calculate : out = \sub a-px-squared py-squared

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Visualizing the vector field

Now we can visualize the vector field using the \vector-field command, which automatically computes the gradient at each point:

• Draw vector field: \vector-field p out

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Reactivity

Try changing the value of a to see how the vector field transforms.

• First we need to create an array for slider values

Notice how the vector field changes shape as you modify a. When a is positive, the vectors point outward in the x-direction, and when a is negative, they point inward.