91Ó°ÊÓ

Partial derivatives are a foundational concept in calculus, specifically in multivariable calculus. They help us understand how a function changes when each of its variables is varied slightly, while all other variables are kept constant. This is crucial when dealing with transformations, like in our exercise.
To find a partial derivative, choose one variable to differentiate with respect to, and treat all other variables as constants. For example, in our transformation equations:

• From the equation \(x = 3u - 4v\), the partial derivative of \(x\) with respect to \(u\) is 3, written as \(\frac{\partial x}{\partial u}\), because differentiating a linear equation in \(u\) gives the coefficient of \(u\).
• Similarly, \(\frac{\partial x}{\partial v} = -4\) because differentiating with respect to \(v\) gives the coefficient of \(v\) in the equation for \(x\).
• For the equation \(y = \frac{1}{2}u + \frac{1}{6}v\), \(\frac{\partial y}{\partial u} = \frac{1}{2}\) and \(\frac{\partial y}{\partial v} = \frac{1}{6}\) using the same process as above.

Computing partial derivatives is essential for setting up the Jacobian matrix, which we will discuss next.

The Jacobian matrix is a construct used to consolidate partial derivatives of multivariable functions into a grid or an array. This is particularly important when transforming variables from one coordinate system to another, like from \((u, v)\) to \((x, y)\).
A Jacobian matrix \(J\) is formed by aligning the partial derivatives of each new variable with respect to all old variables, creating a compact representation of a transformation.
In our exercise:

• For the transformation given by \(x = 3u - 4v\) and \(y = \frac{1}{2}u + \frac{1}{6}v\), the Jacobian matrix \(J\) is constructed as:

\[J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} = \begin{bmatrix} 3 & -4 \ \frac{1}{2} & \frac{1}{6} \end{bmatrix} \]This matrix provides a snapshot of how changes in \(u\) and \(v\) affect \(x\) and \(y\). Understanding the Jacobian matrix is crucial because it not only shows how each variable influences the transformation but also serves as a building block for computing the determinant, which we will explore in the next section.

Calculating the determinant of a Jacobian matrix gives insight into how a transformation affects area scaling between coordinate systems. The determinant tells us whether the transformation preserves orientation and whether there's any scaling involved.
To compute the determinant of a 2x2 matrix like the Jacobian matrix \(J\), use the formula:

• \(\text{det}(J) = a \cdot d - b \cdot c\)

where the matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is made up of the entries:

• \(a = \frac{\partial x}{\partial u} = 3\)
• \(b = \frac{\partial x}{\partial v} = -4\)
• \(c = \frac{\partial y}{\partial u} = \frac{1}{2}\)
• \(d = \frac{\partial y}{\partial v} = \frac{1}{6}\)

Using the formula, compute:\[\text{det}(J) = (3)(\frac{1}{6}) - (-4)(\frac{1}{2}) = \frac{3}{6} + 2 = \frac{1}{2} + 2 = \frac{5}{2}.\]The calculated determinant \(\frac{5}{2}\) indicates that this transformation scales areas by this factor from the original \((u, v)\) coordinates to the \((x, y)\) coordinates. A positive determinant also signifies that the transformation maintains the original orientation.