91Ó°ÊÓ

At its core, the gradient is a multi-dimensional generalization of the derivative, which you may already be familiar with from single-variable calculus. It gives us the direction of steepest ascent of a function of several variables. If you imagine a mountainous landscape, the gradient is like an arrow pointing uphill from any given point.

Mathematically, the gradient of a function \( f(x, y) \) is denoted as \( abla f(x, y) \) and consists of the partial derivatives of \( f \) with respect to each of its variables:

• \( \frac{\partial f}{\partial x} \) - Partial derivative concerning \( x \)
• \( \frac{\partial f}{\partial y} \) - Partial derivative concerning \( y \)

For example, in the original exercise, the gradient of \( f(x, y) = \sin(2x - y) \) at a specific point \( P \) is calculated using the partial derivatives values, yielding \( abla f(P) = (1, -1/2) \). This vector tells us how much and in what direction the function \( f \) increases most rapidly from point \( P \).

A unit vector is a vector with a length of 1. It’s crucial because it simplifies many mathematical computations and allows us to express directions clearly without worrying about the vector's magnitude. When working with directional derivatives, using a unit vector helps to focus solely on the direction's influence on change, free from the scale.

To convert any vector into a unit vector, you divide each component by the vector's magnitude:

• Find the magnitude \( \|\mathbf{a}\| \)
• Divide each vector component by this magnitude: \( \mathbf{u} = \frac{\mathbf{a}}{\|\mathbf{a}\|} \)

In the problem, the vector from \( P \) to \( Q \) must be converted into a unit vector to effectively determine the directional derivative. Here, the vector is \( \overrightarrow{PQ} \) and its unit form becomes \( \mathbf{u} = \left(\frac{2\sqrt{5}}{3}, -\frac{\sqrt{5}}{3}\right) \), ensuring the emphasis remains on its direction.

The rate of change describes how one quantity changes in relation to another. In terms of functions of several variables, it’s about how the function value varies as you move in a particular direction from a given point. The concept is not unlike speed, which is the rate of change of position over time.

In the context of a function \( f \), the rate of change in a specific direction is given by the directional derivative. This directional derivative combines the gradient at the point and the desired direction’s unit vector:

• \( D_{\mathbf{u}} f(P) = abla f(P) \cdot \mathbf{u} \)

Where \( \cdot \) denotes the dot product. It tells us the rate at which \( f \) changes as you travel in direction \( \mathbf{u} \). For example, if you move in the "steepest" direction of \( f \) (which is the gradient), the rate of change is maximized. In specific exercises, like the one above, you find that all changes align with this measure, whether direction maximizes increase or decrease. That is why, at a point \( P\), in the direction of \( (1, -1/2) \), the rate of change is \( \sqrt{5}/2 \), and moves oppositely for decrease.

Partial derivatives are used to understand how a multi-variable function changes as each variable changes while holding the others constant. It's like examining the slope of the function in the direction of one axis at a time and is a building stone for gradients and directional derivatives.

For a function \( f(x, y) \):

• \( \frac{\partial f}{\partial x} \) measures change in \( f \) as \( x \) changes and \( y \) remains constant.
• \( \frac{\partial f}{\partial y} \) measures change in \( f \) as \( y \) changes and \( x \) remains constant.

In our exercise, the partial derivatives of *f* with respect to \( x \) and \( y \) form components of the gradient. For \( f(x, y) = \sin(2x - y) \), these partial derivatives \( 2\cos(2x-y) \) and \(-\cos(2x-y) \) translate the sensitivity of \( f \) to variations in \( x \) and \( y \) respectively. This makes partial derivatives an invaluable tool for grasping the local behavior of complex surfaces.