91Ó°ÊÓ

The gradient is a fundamental concept in multivariable calculus. It allows us to understand how a function changes in different directions.
For a function of two variables, like \(f(x, y)\), the gradient is a vector composed of the function's partial derivatives. The notation \( \text{grad} \, f \) is often used to represent this vector.
To compute the gradient, you calculate each partial derivative separately:

• \( f_x = \frac{\partial f}{\partial x} \), which shows the rate of change of the function with respect to \(x\).
• \( f_y = \frac{\partial f}{\partial y} \), which shows the rate of change of the function with respect to \(y\).

Once computed, these derivatives are placed into a vector:\( \operatorname{grad} f = (f_x, f_y) \). This vector points in the direction of the steepest ascent of the function.
In our example, for the function \(f(x, y) = e^x \cos y\), we determined the gradient as:\( (e^x \cos y, -e^x \sin y) \). At the point \(P = (0, \pi/2)\), the gradient becomes \( (0, -1) \). This tells us that at point \(P\), the function decreases most steeply in the direction of the vector.

A partial derivative is a tool used to understand how a function changes with respect to a single variable, keeping other variables constant. This is crucial in settings where functions depend on more than one variable.
In the function \(f(x, y)\), the partial derivative \(f_x\) signifies how \(f\) changes as \(x\) changes while \(y\) is constant. Similarly, \(f_y\) explains the change in \(f\) when \(y\) changes, keeping \(x\) unchanged. Calculating these partial derivatives involves treating the other variable as a constant:

• To find \(f_x\), differentiate \(f\) with respect to \(x\). In the problem, this resulted in \( e^x \cos y \).
• To find \(f_y\), differentiate \(f\) with respect to \(y\). This gave us \( -e^x \sin y \).

Partial derivatives form the building blocks for the gradient, and therefore they are essential for finding directional derivatives and comprehending the behavior of multivariable functions. This understanding is particularly useful in optimization and real-world applications like physics and engineering.

The dot product is an operation on two vectors that results in a scalar value.
It provides a measure of how closely two vectors align, meaning how much one vector contributes to another. For two vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \), the dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as:\[ a_1b_1 + a_2b_2 \]The directional derivative uses this operation. Specifically, the directional derivative \(D_{\mathbf{u}} f\) is found by computing the dot product between the gradient of a function and a unit vector \( \mathbf{u} \) indicating the direction of interest. In our example, with \( \operatorname{grad} f = (0, -1) \) and \( \mathbf{u} = (0, 1) \), the calculation was:\[ 0 \times 0 + (-1) \times 1 = -1 \]This result tells us the rate at which the function changes at point \(P\) in the direction \( \mathbf{u} \). A negative value, as in our case, indicates a decrease in the function's value in that direction. The dot product thus stands as a bridge connecting vectors and their geometric interpretations in different fields, harmonizing algebra with geometry.