91Ó°ÊÓ

In calculus, the gradient vector is a crucial concept used to understand how a function changes at each point in space. For a function of two variables, such as \( f(x, y) = xy + y^2 \), the gradient vector \( abla f \) is constructed using the partial derivatives. These partial derivatives are the rates of change of the function concerning each of its variables. In our example:

• \( f_x \), the partial derivative with respect to \( x \), is \( y \).
• \( f_y \), the partial derivative with respect to \( y \), is \( x + 2y \).

Thus, the gradient vector in this case is given by \( abla f = (y, x + 2y) \). The gradient provides the direction of the steepest ascent of the function, and its magnitude shows how steep that ascent is. Evaluating the gradient at a specific point, like \( P(3, 2) \), helps in determining the behavior of the function at that instant.

Orthogonality is a concept used to determine if two vectors are perpendicular to each other. Two vectors are orthogonal if their dot product is zero. In the context of directional derivatives, orthogonality plays a crucial role because if the gradient vector \( abla f \) is orthogonal to a direction vector \( \mathbf{v} \), it means that moving in the direction of \( \mathbf{v} \) does not increase or decrease the function value. This results in the directional derivative being zero.To determine orthogonality between the gradient vector and the direction vector \( \mathbf{v} = (a, b) \) at point \( P(3, 2) \), we compute the dot product:

• \( abla f(3, 2) = (2, 7) \).
• The dot product with the direction vector is \( 2a + 7b = 0 \).

Thus, any direction vector \( (a, b) \) that satisfies this equation, such as \( (-7, 2) \), is orthogonal to the gradient vector, ensuring that the directional derivative is zero.

Partial derivatives are a fundamental part of studying multivariable functions. They capture how a function changes with respect to one variable while keeping the other variables constant. For the function \( f(x, y) = xy + y^2 \),

• The partial derivative \( f_x = y \) implies that when you change \( x \) slightly, the function's change is directly proportional to \( y \).
• Similarly, \( f_y = x + 2y \) indicates that the change in the function as \( y \) changes depends on both \( x \) and \( y \).

These derivatives are used to form the gradient vector, which helps understand the direction of maximal change of the function. By evaluating these derivatives at specific points, you gain insights into the local behavior of the function, essential for applications such as optimization and finding zero directional derivatives.