Problem 8 - Advanced Engineering Mathematics

Question

Find the Jacobian of the transformation \(T\) from the \(u v\)-plane to the \(x y\)-plane. $$ x=e^{3 u} \sin v, y=e^{3 u} \cos v $$

Step-by-Step Solution

Answer

The Jacobian determinant is 3 \(e^{6u}\).

Step 1: Identify Partial Derivatives

To find the Jacobian, we need the partial derivatives of \(x\) and \(y\) with respect to \(u\) and \(v\). This will form the Jacobian matrix \(J\).Start by calculating:1. \( \frac{\partial x}{\partial u} = \frac{\partial}{\partial u} (e^{3u} \sin v) = 3e^{3u} \sin v \)2. \( \frac{\partial x}{\partial v} = \frac{\partial}{\partial v} (e^{3u} \sin v) = e^{3u} \cos v \)3. \( \frac{\partial y}{\partial u} = \frac{\partial}{\partial u} (e^{3u} \cos v) = 3e^{3u} \cos v \)4. \( \frac{\partial y}{\partial v} = \frac{\partial}{\partial v} (e^{3u} \cos v) = -e^{3u} \sin v \)

Key Concepts

Partial Derivatives

Partial derivatives are crucial in understanding how variables change in a multivariable system. In simpler terms, they help us see how a function responds when we change just one of its input variables, keeping the others constant.

Let's break it down with our transformation exercise. We have the equations:

• For \(x = e^{3u} \sin v\), the partial derivative \(\frac{\partial x}{\partial u}\) focuses on how \(x\) changes as \(u\) changes, when \(v\) is constant.
• Similarly, \(\frac{\partial x}{\partial v}\) looks at how \(x\) changes with \(v\) varying, with \(u\) constant.

The same logic applies for \(y = e^{3u} \cos v\). Calculating these allows us to build our next concept: the Jacobian Matrix.

Transformation

A transformation in mathematics is like translating a story from one language to another. It's about changing coordinates from one system, like the \(uv\)-plane, to another, such as the \(xy\)-plane.

In our exercise, this involves moving from inputs \(u\) and \(v\) to outputs \(x\) and \(y\) using specific functions:

• \(x = e^{3u} \sin v\)
• \(y = e^{3u} \cos v\)

These transformations reshape space, and understanding them is key to solving the problem at hand. By calculating these, you can understand how every point in the \(uv\)-plane moves to a new place in the \(xy\)-plane.

Jacobian Matrix

The Jacobian Matrix is a powerful tool in multivariable calculus, like a map that guides us through transformations.

For our exercise, it consists of the partial derivatives we calculated:

• \(\frac{\partial x}{\partial u} = 3e^{3u} \sin v\)
• \(\frac{\partial x}{\partial v} = e^{3u} \cos v\)
• \(\frac{\partial y}{\partial u} = 3e^{3u} \cos v\)
• \(\frac{\partial y}{\partial v} = -e^{3u} \sin v\)

Arranged in a matrix form, it looks like this:\[J = \begin{bmatrix} 3e^{3u} \sin v & e^{3u} \cos v \3e^{3u} \cos v & -e^{3u} \sin v\end{bmatrix}\]This Jacobian matrix helps in understanding the rate of change of the transformation and detecting important properties like inversion or distortion of space.

Multivariable Calculus

Multivariable calculus extends the basic concepts of calculus to more than one variable. This is essential for tackling the complex systems we see not only in mathematics, but also in fields like physics and engineering.

The exercise involves both transformation and partial derivatives, illustrating how they contribute to forming the Jacobian. In multivariable calculus:

• Partial derivatives help us understand changes in one variable at a time, revealing how each influences the overall system.
• Transformations show how points move and change shape across different coordinate systems.
• The Jacobian matrix combines these concepts, giving a clear tool to evaluate how transformations affect areas and shapes.

Understanding these concepts clarifies how different aspects of a problem interact, offering deep insights into the nature of multivariable fields.