91Ó°ÊÓ

In multivariable calculus, a composite function involves combining two or more functions such that the output of one becomes the input of another. For the exercise above, the composite function is defined as \( h(x, y) = f(\mathbf{g}(x, y)) \). This means that first, we apply the mapping \( \mathbf{g} \) to the variables \( (x, y) \), resulting in a new set of variables \((u, v)\).

• Here, \( \mathbf{g}(x, y) = (xy, x + 2y)^T \), a transformation that combines and changes the input pair.
• Then, these transformed values are fed into the function \( f \), making the composite function \( h(x, y) \), which essentially alters inputs through two layers.

This composition is crucial in analyzing problems where direct input-output relationships are complex, easing the overall computation by breaking down into manageable parts.

The gradient is a fundamental concept in calculus that indicates the direction and rate of increase of a function. For a scalar function like \( f \), the gradient \( abla f \) is a vector of its partial derivatives. This vector points in the direction of the greatest rate of increase of the function at a given point.
In the exercise, the gradient of \( f \) is calculated in terms of the transformed variables \( (u, v) \) from the function \( g(x, y) \). It is expressed as \( [\frac{\partial f}{\partial u}, \frac{\partial f}{\partial v}]^T \), and in this case becomes \([\sin(u^2)v, \sin(u^2)]^T\) after substituting the expressions for \( f' \).

• For a practical approach, evaluating the gradient entails substituting specific values of \( (u, v) \) to assess how \( f \) changes around that point.
• The direction and magnitude indicated by the gradient are pivotal in finding local maximums and minimums in real-world applications.

Gradient vectors are notable for pointing towards increasing values of functions, which is central to optimization and related computations.

The Jacobian matrix is an essential tool for representing the linear approximation of functions, particularly in transformations. It comprises partial derivatives and provides insights into the behavior of multi-variable functions.
For the mapping \( \mathbf{g}(x, y) = (xy, x + 2y)^T \), the Jacobian matrix \( J_g \) is calculated as follows:

• The first row includes the partial derivatives of the first component \( xy \) with respect to \( x \) and \( y \): \( [y, x] \).
• The second row includes the partial derivatives of the second component \( x + 2y \): \( [1, 2] \).

Therefore, the Jacobian matrix is:\[J_g = \begin{pmatrix} y & x \ 1 & 2 \end{pmatrix}\]Evaluating this matrix at \( (1, 0) \) yields:\[J_g(1, 0) = \begin{pmatrix} 0 & 1 \ 1 & 2 \end{pmatrix}\]This matrix helps in transforming the gradient and understanding how small changes in inputs at a specific point linearly affect the outcomes.

The chain rule is a method in calculus used for computing the derivative of composite functions. It is vital when a variable depends on other variables indirectly through other functions.
The exercise uses the chain rule to find the gradient of the composite function \( h(x, y) = f(\mathbf{g}(x, y)) \). The chain rule here states that:

• First, compute the gradient of \( f \) with respect to the variables it directly depends on, \( (u, v) \).
• Then, multiply this gradient by the transpose of the Jacobian matrix \( J_g \) of the transformation \( g \).

In mathematical terms, it involves finding \( abla h(x, y) = J_g(x, y)^T abla f(g(x, y)) \). For the point \( (1, 0) \), we apply this formula, yielding a zero gradient vector: \[(J_g(1, 0))^T \cdot \begin{pmatrix} 0 \ 0 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \end{pmatrix}\]This result indicates that there is no direction of steepest ascent, showcasing the chain rule's efficacy in unfolding the relationship between multi-variable transformations and their impacts. This mathematical synergy is crucial in both theoretical derivations and practical applications.