Problem 8 - Advanced Engineering Mathematics

Question

In Problems \(7-10\), 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 of the transformation is \\(-3e^{6u}\\).

Step 1: Recall Definition of Jacobian

The Jacobian matrix relates changes in variables from one coordinate system to another. For a transformation from \(u, v\) to \(x, y\), the Jacobian matrix \((J)\) is defined as: \[ J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} \] To find the Jacobian determinant, we need to compute these partial derivatives.

Step 2: Compute Partial Derivatives

First, compute the partial derivatives for \(x = e^{3u} \sin v\):- \(\frac{\partial x}{\partial u} = 3e^{3u} \sin v\)- \(\frac{\partial x}{\partial v} = e^{3u} \cos v\)Next, for \(y = e^{3u} \cos v\):- \(\frac{\partial y}{\partial u} = 3e^{3u} \cos v\)- \(\frac{\partial y}{\partial v} = -e^{3u} \sin v\)

Step 3: Form the Jacobian Matrix

Substitute the computed partial derivatives into the Jacobian matrix: \[ J = \begin{bmatrix} 3e^{3u} \sin v & e^{3u} \cos v \ \ 3e^{3u} \cos v & -e^{3u} \sin v \end{bmatrix} \]

Step 4: Calculate the Jacobian Determinant

The Jacobian determinant is found as follows:\[ det(J) = \left(3e^{3u} \sin v\right)(-e^{3u} \sin v) - \left(e^{3u} \cos v\right)(3e^{3u} \cos v) \]Simplifying gives:\[ det(J) = -3e^{6u} \sin^2 v - 3e^{6u} \cos^2 v \]Using the identity \(\sin^2 v + \cos^2 v = 1\), we find:\[ det(J) = -3e^{6u} \times 1 = -3e^{6u} \]

Step 5: Conclude the Result

The Jacobian determinant, which describes how the area changes under the transformation from \(u, v\) to \(x, y\), is found to be \(-3e^{6u}\).

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental tool in calculus, especially useful when working with functions of several variables. They measure how a function changes as one of its input variables changes, keeping the other variables constant. Imagine you are climbing a hill, partial derivatives tell you how steep the hill is in a specific direction like north or east, without changing your east or north direction respectively.

When given a function, such as \(x = e^{3u} \sin v\) or \(y = e^{3u} \cos v\), partial derivatives help us understand how \(x\) or \(y\) changes as \(u\) or \(v\) changes. Here's a simple breakdown of partial derivatives involved in our transformation:

• \(\frac{\partial x}{\partial u} = 3e^{3u} \sin v\): This tells us how \(x\) changes as \(u\) changes, while \(v\) is kept constant.
• \(\frac{\partial x}{\partial v} = e^{3u} \cos v\): This tells us how \(x\) changes as \(v\) changes, while \(u\) is kept fixed.
• Similarly, \(\frac{\partial y}{\partial u}\) and \(\frac{\partial y}{\partial v}\) show how \(y\) changes with respect to \(u\) and \(v\).

Computing these derivatives allows us to form the Jacobian matrix, which highlights how the entire system behaves in response to small changes in \(u\) and \(v\).

Coordinate Transformation

Coordinate transformation is a method used in mathematics to change the basis of a coordinate system. Think of it as converting a position from one map to another. In our problem, we convert coordinates from the \(uv\)-plane to the \(xy\)-plane.

By transforming coordinates, we can express objects and quantities in a new frame of reference, which is often beneficial for easier calculation or better understanding. Here, the transformation is accomplished through:

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

This transformation takes points defined by \((u, v)\) and maps them onto new points \((x, y)\). The new coordinates describe the same point but in a possibly more convenient or insightful way.

Understanding coordinate transformation is crucial in fields like physics, engineering, and computer graphics, where different perspectives and representations are needed for analysis. It's important to note that this process keeps the underlying relationships intact, even though the numerical values of those points change.

Jacobian Determinant

The Jacobian determinant is a powerful concept in multi-variable calculus, used to understand how an area, volume, or higher-dimensional space changes under transformation. In simple terms, it describes how a tiny square area in the \(uv\)-plane will distort into a new shape in the \(xy\)-plane.

For our exercise, the Jacobian matrix was constructed from partial derivatives of the transformation functions. To find the Jacobian determinant, we calculate:
\[\text{det}(J) = \begin{vmatrix} 3e^{3u} \sin v & e^{3u} \cos v \ 3e^{3u} \cos v & -e^{3u} \sin v \end{vmatrix}\]Solving this determinant gives us \[-3e^{6u}\], which succinctly describes the scaling factor for areas.

If the Jacobian determinant is positive, the orientation is preserved, while a negative value indicates a flip or reversal in orientation. Here, the negative sign means the object's orientation flips when transformed from the \(uv\) plane to the \(xy\) plane.

Therefore, the Jacobian determinant not only measures how much an object stretches or compresses but also gives information about its orientation in a new coordinate system.