91Ó°ÊÓ

The gradient vector is a fundamental concept in calculus, especially when dealing with multivariable functions. Think of it as a tool that tells us the direction in which a function is increasing most steeply. For a function of two variables, say \( f(x, y) \), the gradient is denoted as \( abla f(x, y) \) and is a vector composed of the partial derivatives of \( f \) with respect to \( x \) and \( y \). Specifically, the gradient vector can be expressed as:

• \( \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \)

In our exercise, the gradient vector for \( f(x, y) = 2x^2 + 3xy + 4y^2 \) was calculated to be \( abla f(x, y) = (4x + 3y, 3x + 8y) \). At point \( P(1, 1) \), it becomes \( (7, 11) \). This vector not only points out the direction of the greatest rate of change of the function but its magnitude also provides the steepest slope of increase.

A partial derivative represents the rate at which a function changes as one of the variables changes, while holding the other variable constant. This is a cornerstone concept in multivariable calculus, as it helps decompose changes in multivariable functions into simpler single-variable changes.

• For \( f(x, y) = 2x^2 + 3xy + 4y^2 \), the partial derivative with respect to \( x \) is calculated by differentiating \( f \) treating \( y \) as a constant, resulting in \( \frac{\partial f}{\partial x} = 4x + 3y \).
• Similarly, the partial derivative with respect to \( y \) treats \( x \) as a constant, yielding \( \frac{\partial f}{\partial y} = 3x + 8y \).

Partial derivatives are typically used to build the gradient vector, and they give insight into how the function behaves as each variable changes independently.

The magnitude of a vector is akin to the length of the vector, giving a sense of its size but not its direction. To calculate the magnitude, one applies the Pythagorean theorem in multiple dimensions. The formula in a two-dimensional space for a vector \( \mathbf{v} = (a, b) \) is:

• \( \| \mathbf{v} \| = \sqrt{a^2 + b^2} \)

In the context of our gradient vector \( abla f(1, 1) = (7, 11) \), its magnitude gives us the maximum rate of increase of the function at the point \( P(1, 1) \). Using the formula, we find \( \| abla f(1, 1) \| = \sqrt{49 + 121} = \sqrt{170} \). The magnitude is crucial because it quantifies how fast the function is increasing in the direction of the gradient vector.

A unit vector is a vector that has a magnitude of exactly 1. It is often used to specify a direction without concern for magnitude. To convert a vector into a unit vector that points in the same direction, you divide the vector by its magnitude. If \( \mathbf{v} = (a, b) \), the unit vector \( \mathbf{u} \) in the same direction can be calculated as:

• \( \mathbf{u} = \frac{1}{\sqrt{a^2 + b^2}} (a, b) \)

For our gradient vector \( abla f(1, 1) = (7, 11) \), the corresponding unit vector is:

• \( \mathbf{u} = \frac{1}{\sqrt{170}} (7, 11) \)

The unit vector serves to express the direction of the maximum directional derivative without scaling it by its magnitude.