# Gradient Determination of Functions

Calculate the gradient Vf of the following functions, f(x,y,z)

a. f = x^2 + z^3

b. f = ky where k is a constant

c. f = r = (x^2 + y^2 + z^2)^1/2 Hint use the chain rule

d. f = 1/r

See attachment for better symbol representation

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The Del operator is a vector operator. This means that it has three components (in three dimensional space) and it operates on a scalar function:

When applied to a scalar function this becomes:

So, this way we gat a vector from a scalar function.

For example, the potential energy is scalar function (it has no direction). The force applied on a particle moving under the influence of this potential is defined as the negative of the gradient of the energy. The force is a vector.

Since the Del operator is a vector, we can use it like any other vector in vector arithmetic.

For example, the general vector field F is:

Where u,v and w are scalar functions.

So we can use the vector dot product:

This is called the divergence, and the result is a scalar function.

And also the vector cross product:

This operation is called the curl (or "rotor") of F and it results in a vector perpendicular to the field F. This reason for the name is that in many applications it is an indication to the "curviness" of the vector field.

Back to our problem of the gradient:

Let's recall what is the derivative of where a is a constant.

Then:

We get the radius unit vector.

Let's recall what is the derivative of where a is a constant.

Then:

The last result shows that any potential that goes like 1/r generates a force that behave like the square of the distance - most importantly the electrostatic and gravitational forces.

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