Question

Evaluate the Jacobians \(J(u, v, w)\) for the following transformations. $$x=v w, y=u w, z=u^{2}-v^{2}$$

Short answer

Answer: The Jacobian matrix for the given transformation is: $$ J(u,v,w) = \begin{bmatrix} 0 & w & v \\ w & 0 & u \\ 2u & -2v & 0 \end{bmatrix} $$

Step-by-step solution

Calculate the partial derivative of x with respect to u, v, and w

Recall the transformation: \(x = vw\). To find the partial derivatives, we'll take the derivative of x with respect to u, v, and w: $$ \frac{\partial x}{\partial u} = 0 \\ \frac{\partial x}{\partial v} = w \\ \frac{\partial x}{\partial w} = v $$

Calculate the partial derivative of y with respect to u, v, and w

Recall the transformation: \(y = uw\). To find the partial derivatives, we'll take the derivative of y with respect to u, v, and w: $$ \frac{\partial y}{\partial u} = w \\ \frac{\partial y}{\partial v} = 0 \\ \frac{\partial y}{\partial w} = u $$

Calculate the partial derivative of z with respect to u, v, and w

Recall the transformation: \(z = u^2 - v^2\). To find the partial derivatives, we'll take the derivative of z with respect to u, v, and w: $$ \frac{\partial z}{\partial u} = 2u \\ \frac{\partial z}{\partial v} = -2v \\ \frac{\partial z}{\partial w} = 0 $$

Write the Jacobian matrix

Now that we have all of the partial derivatives, we can write the Jacobian matrix \(J(u,v,w)\): $$ J(u,v,w) = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{bmatrix} = \begin{bmatrix} 0 & w & v \\ w & 0 & u \\ 2u & -2v & 0 \end{bmatrix} $$ The Jacobian for the given transformation is: $$ J(u,v,w) = \begin{bmatrix} 0 & w & v \\ w & 0 & u \\ 2u & -2v & 0 \end{bmatrix} $$

Key concepts

Partial Derivatives

In the realm of calculus, especially when dealing with functions of several variables, partial derivatives play a pivotal role. Simply put, a partial derivative measures how a function changes as just one of the variables is varied, with all others held constant.

• For example, when we computed the derivative of the expression \(x = vw\), we isolated each variable (\(u\), \(v\), and \(w\)) to see how changes in each affect \(x\).
• Partial derivatives are denoted by \(\frac{\partial}{\partial x}\), which is similar to the notation for ordinary derivatives but with round \(\partial\) symbols that indicate partial differentiation.

Mastering partial derivatives is essential for analyzing multidimensional functions, as they are fundamental in constructing the Jacobian matrix.

Transformation

Transformation in mathematics often refers to a change of variables to describe a function or a set of functions in a different coordinate system. In the given problem, we consider the mappings \(x = vw\), \(y = uw\), and \(z = u^2 - v^2\) as a transformation.

• These transformations redefine the coordinate system from original variables \((x, y, z)\) to a new set \((u, v, w)\).
• This involves using the transformed coordinates to express how changes in \(x, y,\) and \(z\) relate to each other, which is fundamental in analyzing complex systems.

The Jacobian matrix captures this relationship, describing the sensitivity of the transformed system's output to changes in its input variables. Such transformations are crucial in applications ranging from computer graphics to physics simulations.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of multiple variables. Unlike single-variable calculus, it enables the exploration of geometric, physical, and theoretical spaces with several dimensions.

• In multivariable calculus, we deal with functions that have inputs and outputs across more than one dimension like the "\(u, v, w\)" coordinates in our transformation problem.
• Key concepts include partial derivatives, gradients, and the Jacobian, all of which help to analyze how functions behave in a multi-dimensional space.

Understanding multivariable calculus is essential for fields such as engineering, physics, and economics, where many practical problems naturally result from multi-dimensional processes.