91Ó°ÊÓ

In the study of functions with multiple variables, the **Gradient** is a vector that points in the direction of the greatest rate of increase of the function. Think of it as a multi-dimensional generalization of a derivative. For any point \(x, y, z\) in a function \(f(x, y, z)\), the gradient gives us valuable information about the slope of the function at that point.
- The gradient is composed of partial derivatives with respect to each variable.
- It often looks like \( abla f(x, y, z) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \).
For our given problem, where \(f(x, y, z) = e^{x} \cos yz\), the partial derivatives are:
- \( \frac{\partial f}{\partial x} = e^x \cos(yz) \)
- \( \frac{\partial f}{\partial y} = -e^x z \sin(yz) \)
- \( \frac{\partial f}{\partial z} = -e^x y \sin(yz) \)
The gradient vector here is thus \( \left( e^x \cos(yz), -e^x z \sin(yz), -e^x y \sin(yz) \right) \). At the origin, this simplifies to \( (1, 0, 0) \). This means movement in the x-direction at the origin results in the greatest increase in function value.

A **Unit Vector** is a vector with a magnitude (or length) of one. These vectors retain directional information without changing the original direction's essence. They are crucial when dealing with directions, as they provide a normalized direction vector which is used in calculations like the directional derivative.
To convert a given vector into a unit vector, follow these steps:

• Calculate the magnitude of the vector.
• Divide each component of the vector by this magnitude.

In our exercise, the direction vector given is \( \mathbf{v} = 2\mathbf{i} + 2\mathbf{j} - 2\mathbf{k} \). To find its magnitude, we calculate:
\[|| \mathbf{v} || = \sqrt{2^2 + 2^2 + (-2)^2} = \sqrt{12} = 2\sqrt{3}\]This makes the unit vector \( \mathbf{u} = \frac{1}{2\sqrt{3}}(2\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}) \), which simplifies to \( \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right) \).
Using unit vectors ensures we focus purely on direction without being skewed by magnitude.

**Partial Derivatives** play a fundamental role in functions having multiple variables, similar to regular derivatives but focusing on a change in one variable while keeping others constant. They help to understand how the function behaves when one variable changes slightly.
For a function \( f(x, y, z) \), partial derivatives are written as \( \frac{\partial f}{\partial x} \, \, \frac{\partial f}{\partial y} \, \text{and} \, \frac{\partial f}{\partial z} \). Each shows the rate of change of the function when the respective variable changes.
In our problem, the partial derivatives were calculated as follows:

• \(\frac{\partial f}{\partial x} = e^{x} \cos(yz)\)
• \(\frac{\partial f}{\partial y} = -e^{x}z \sin(yz)\)
• \(\frac{\partial f}{\partial z} = -e^{x}y \sin(yz)\)

At the origin, these derivatives help form the gradient, showing the direction of steepest ascent. Understanding these derivatives is crucial for building the gradient and deriving directional derivatives.

The **Change in Function** represents how much the value of the function is expected to increase or decrease when moving a small distance in a specified direction. The calculation usually involves the directional derivative.
The **Directional Derivative** identifies this rate of change in any given direction by taking a dot product of the gradient and a unit vector in that direction.
In our scenario, the directional derivative at the origin was calculated as:
\[D_\mathbf{u}f(0,0,0) = abla f(0,0,0) \cdot \mathbf{u} = \left( 1, 0, 0 \right) \cdot \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right) = \frac{1}{\sqrt{3}}\]Finally, to find the change in function as the point moves, multiply the directional derivative by the distance traveled:
\[\Delta f \approx D_\mathbf{u}f(0,0,0) \times ds = \frac{1}{\sqrt{3}} \times 0.1 = \frac{0.1}{\sqrt{3}} \approx 0.0577\]This approximates how much the function changes when you move from the origin in the specified direction.