91Ó°ÊÓ

The gradient is a crucial concept in multivariable calculus. It tells us how a function changes at any given point. For a function of two variables like \(f(x, y)\), the gradient vector is composed of its partial derivatives.
For the function \(f(x, y) = x^2 - y^2\), the gradient is derived from the partial derivatives:

• \(\frac{\partial f}{\partial x} = 2x\) - this is the rate of change of \(f\) as \(x\) changes while keeping \(y\) constant.
• \(\frac{\partial f}{\partial y} = -2y\) - this is the rate of change of \(f\) as \(y\) changes while keeping \(x\) constant.

The gradient \(abla f = (2x, -2y)\) points in the direction of the greatest increase of the function. It plays a critical role in understanding how vector fields behave within the plane, providing insights into the nature of the function itself.

The tangent vector is essentially a directional vector that touches a curve at a specific point. It's crucial for understanding the behavior of a path in a plane or space. In this problem, the tangent vector \(\mathbf{T}\) has its roots in the velocity vector \(\mathbf{v}\).
Given the expressions \(x = x_0 + 2t\) and \(y = y_0 + 3t\), the velocity vector \(\mathbf{v} = (2, 3)\) is derived by differentiating these with respect to \(t\), which involves the motion equations.
The tangent vector \(\mathbf{T}\) is obtained by normalizing \(\mathbf{v}\) (dividing by its magnitude):

• Magnitude of \(\mathbf{v}\) = \(\sqrt{2^2 + 3^2} = \sqrt{13}\)
• \(\mathbf{T} = \left( \frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}} \right)\)

Thus, \(\mathbf{T}\) provides the unit direction along the curve at any point. Understanding tangent vectors helps in visualizing how a curve "flows" in space around a specific point.

The rate of change in the context of functions of several variables is often handled through directional derivatives. The expression \(\frac{df}{dt} = \operatorname{grad} f \cdot \mathbf{v}\) calculates how the function \(f(x, y)\) changes as you move along the trajectory defined by \(\mathbf{v}\).
This involves a dot product of the gradient with the velocity vector:

• Gradient: \((2x, -2y)\)
• Velocity vector \(\mathbf{v} = (2, 3)\)
• Dot product yields: \(\operatorname{grad} f \cdot \mathbf{v} = 4x - 6y\)

This represents the instantaneous rate of change of \(f\) at a point, as you move with the velocity \(\mathbf{v}\). It's pivotal in many fields including physics and engineering, illustrating how rapidly a quantity may be changing along a given path.

In multivariable functions, partial derivatives provide insights on how a function changes with respect to one variable, while keeping others constant. This is important for understanding local behaviors of functions.
For \(f(x, y) = x^2 - y^2\), we calculate the partial derivatives:

• \(\frac{\partial f}{\partial x} = 2x\) - indicates how \(f\) changes as \(x\) increases while \(y\) remains the same.
• \(\frac{\partial f}{\partial y} = -2y\) - shows the change in \(f\) as \(y\) increases while \(x\) stays constant.

Partial derivatives help form the gradient vector, acting as fundamental tools in analyzing changes from various inputs. They are integral in contour mapping and optimizations, providing the foundation for more complex derivative operations in calculus.