91Ó°ÊÓ

The gradient of a function is like a multi-dimensional version of the derivative. Essentially, it indicates the direction and rate of fastest increase of a function.
When dealing with a function of several variables, its gradient is the vector formed by the collection of its partial derivatives.
For our example, with the function

• \( f(x, y) = x^2 - y^2 \),

the gradient is expressed as \( abla f = (2x, -2y) \).
This means:

• \( 2x \) shows how much \( f \) changes as \( x \) changes,
• \(-2y \) shows how \( f \) changes as \( y \) changes.

By evaluating this gradient at a specific point \( P(1, 0) \), we determine how steeply the function values change at that specific location, resulting in the vector \((2, 0)\).

Partial derivatives are a fundamental tool in calculus for dealing with functions of multiple variables. They capture how a function changes as one of the variables changes, while keeping the other variables constant.
For a function \( f(x, y) = x^2 - y^2 \), the partial derivative with respect to \( x \) is calculated by treating \( y \) as a constant, resulting in \( \frac{\partial f}{\partial x} = 2x \). Similarly, for the partial derivative with respect to \( y \), \( x \) is treated as constant, leading to \( \frac{\partial f}{\partial y} = -2y \).

• Partial derivatives help us derive the Gradient vector.
• They are key to understanding how each independent variable contributes to the behavior of the function.

By knowing these derivatives, the gradient of the function becomes intuitive as it combines all these changes into a single vector.

A unit vector is a vector with a magnitude of 1. It is often used to specify a direction without any concern for length or size.
To ensure a vector is a unit vector, compute its magnitude and confirm this equals 1.
The unit vector \( \mathbf{u} = \left\langle \frac{\sqrt{3}}{2}, \frac{1}{2} \right\rangle \) in our problem was verified by calculating:

• \( \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2} = 1 \).

Using the unit vector ensures that the directional derivative expresses the rate and direction of change, rather than the overall amount of change, providing a normalized direction for evaluating how the function behaves along that vector.

The dot product of two vectors provides a scalar value that reflects the extent to which one vector extends in the direction of another.
It is a crucial operation in computing directional derivatives.
Mathematically, the dot product of two vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \) is determined as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).

For the gradient vector \( abla f = (2, 0) \) and the unit vector \( \mathbf{u} = \left\langle \frac{\sqrt{3}}{2}, \frac{1}{2} \right\rangle \), their dot product calculates the directional derivative:

• \((2) \cdot \left(\frac{\sqrt{3}}{2}\right) + (0) \cdot \left(\frac{1}{2}\right) = \sqrt{3} \).

This result represents the rate of change of the function \( f(x, y) \) at point \( P(1, 0) \) in the direction specified by the unit vector.