91Ó°ÊÓ

In understanding directional derivatives, the gradient vector is a crucial concept. Imagine it as a compass that guides you in the direction where a function's increase is the steepest. For a function of several variables, the gradient is a vector formed by the partial derivatives with respect to each variable involved.
The gradient of a function \( f(x, y, z) = x^2 + 2y^2 - 3z^2 \) is given as:

• The partial derivative with respect to \( x \) is \( \frac{\partial f}{\partial x} = 2x \).
• The partial derivative with respect to \( y \) is \( \frac{\partial f}{\partial y} = 4y \).
• The partial derivative with respect to \( z \) is \( \frac{\partial f}{\partial z} = -6z \).

Thus, the gradient vector \( abla f = (2x, 4y, -6z) \).
To evaluate it at a specific point such as \( P_0(1, 1, 1) \), you simply substitute the coordinates of that point into the gradient, yielding the vector \( (2, 4, -6) \). This vector not only tells you the direction of steepest ascent but also its rate.

Partial derivatives are the foundation blocks for constructing the gradient vector. They signify how a function changes as you alter one variable, keeping others constant. Imagine a three-dimensional graph. A partial derivative examines the slope of the function along one plane of this graph while fixing other variables.
For the function \( f(x, y, z) = x^2 + 2y^2 - 3z^2 \), each partial derivative is computed as follows:

• \( \frac{\partial f}{\partial x} = 2x \) shows how the function changes with \( x \), assuming \( y \) and \( z \) are fixed.
• \( \frac{\partial f}{\partial y} = 4y \) indicates the change with \( y \), keeping \( x \) and \( z \) constant.
• \( \frac{\partial f}{\partial z} = -6z \) represents change with \( z \), with \( x \) and \( y \) unchanged.

These derivatives are organized into the gradient vector, conveying multi-directional change in the function's value.

Unit vectors are essential when determining directional derivatives as they preserve the direction while standardizing the magnitude to one. A direction vector, such as \( \mathbf{u} = \mathbf{i} + \mathbf{j} + \mathbf{k} \), is expressed in terms of its components along each axis.
However, to use it in computations like the directional derivative, it must be normalized (converted to a unit vector). For the vector \( (1, 1, 1) \), compute its magnitude:

• \( |\mathbf{u}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \)

The unit vector is then given by dividing each component by the magnitude:

• \( \hat{\mathbf{u}} = \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \)

Using unit vectors ensures uniformity in calculations and makes it possible to assess changes in any specified direction effectively. The dot product of the unit vector with the gradient provides the directional derivative, simplifying the influence of scale.