Question

Meaning of the Jacobian The Jacobian is a magnification (or reduction) factor that relates the area of a small region near the point \((u, v)\) to the area of the image of that region near the point \((x, y)\) a. Suppose \(S\) is a rectangle in the \(u v\) -plane with vertices \(O(0,0)\) \(P(\Delta u, 0),(\Delta u, \Delta v),\) and \(Q(0, \Delta v)\) (see figure). The image of \(S\) under the transformation \(x=g(u, v), y=h(u, v)\) is a region \(R\) in the \(x y\) -plane. Let \(O^{\prime}, P^{\prime},\) and \(Q^{\prime}\) be the images of O, \(P,\) and \(Q,\) respectively, in the \(x y\) -plane, where \(O^{\prime}, P^{\prime},\) and \(Q^{\prime}\) do not all lie on the same line. Explain why the coordinates of \(\boldsymbol{O}^{\prime}, \boldsymbol{P}^{\prime},\) and \(Q^{\prime}\) are \((g(0,0), h(0,0)),(g(\Delta u, 0), h(\Delta u, 0))\) and \((g(0, \Delta v), h(0, \Delta v)),\) respectively. b. Use a Taylor series in both variables to show that $$\begin{array}{l} g(\Delta u, 0) \approx g(0,0)+g_{u}(0,0) \Delta u \\ g(0, \Delta v) \approx g(0,0)+g_{v}(0,0) \Delta v \\ h(\Delta u, 0) \approx h(0,0)+h_{u}(0,0) \Delta u \\ h(0, \Delta v) \approx h(0,0)+h_{v}(0,0) \Delta v \end{array}$$ where \(g_{u}(0,0)\) is \(\frac{\partial x}{\partial u}\) evaluated at \((0,0),\) with similar meanings for \(g_{v}, h_{u},\) and \(h_{v}\) c. Consider the vectors \(\overrightarrow{O^{\prime} P^{\prime}}\) and \(\overrightarrow{O^{\prime} Q^{\prime}}\) and the parallelogram, two of whose sides are \(\overrightarrow{O^{\prime} P^{\prime}}\) and \(\overrightarrow{O^{\prime} Q^{\prime}}\). Use the cross product to show that the area of the parallelogram is approximately \(|J(u, v)| \Delta u \Delta v\) d. Explain why the ratio of the area of \(R\) to the area of \(S\) is approximately \(|J(u, v)|\)

Short answer

Answer: The resulting ratio is approximately |J(u,v)|, where J(u,v) is the Jacobian evaluated at the point (u,v).

Step-by-step solution

Coordinates of Transformed Points

We are given a transformation \(x = g(u,v), y = h(u,v)\). To find the coordinates of the transformed points \(O'\), \(P'\), and \(Q'\), we simply substitute the coordinates of the original points \(O\), \(P\), and \(Q\) into the transformation equations. For \(O'\), we have \((x,y) = (g(0,0), h(0,0))\). For \(P'\), we have \((x,y) = (g(\Delta u, 0), h(\Delta u, 0))\). For \(Q'\), we have \((x,y) = (g(0, \Delta v), h(0, \Delta v))\).

Taylor Series Calculation

Now, we'll approximate \(g(\Delta u, 0)\), \(g(0, \Delta v)\), \(h(\Delta u, 0)\), and \(h(0, \Delta v)\) using Taylor series. For each function, we will do a first-order (linear) approximation, with error terms being proportional to \(\Delta u^2\) and \(\Delta v^2\). This gives us the expressions stated in the exercise: \(g(\Delta u, 0) \approx g(0,0)+g_{u}(0,0) \Delta u\), \(g(0, \Delta v) \approx g(0,0)+g_{v}(0,0) \Delta v\), \(h(\Delta u, 0) \approx h(0,0)+h_{u}(0,0) \Delta u\), \(h(0, \Delta v) \approx h(0,0)+h_{v}(0,0) \Delta v\).

Area of the Parallelogram

We want to find the area of the parallelogram formed by the vectors \(\overrightarrow{O'P'}\) and \(\overrightarrow{O'Q'}\). Using the cross product of these vectors, we get: \(\overrightarrow{O'P'} = \begin{bmatrix}g_u(0,0)\Delta u \\ h_u(0,0)\Delta u\end{bmatrix}\), and \(\overrightarrow{O'Q'} = \begin{bmatrix}g_v(0,0)\Delta v \\ h_v(0,0)\Delta v\end{bmatrix}\). Taking the cross product and finding its magnitude, we get the area of the parallelogram: \(Area = \left|\begin{bmatrix} g_u(0,0)~\Delta u \\ h_u(0,0)~\Delta u \end{bmatrix}~\times~\begin{bmatrix} g_v(0,0)~\Delta v \\ h_v(0,0)~\Delta v \end{bmatrix}\right| = |J(u, v)| \Delta u \Delta v\).

Area Ratio

Finally, we want to find the ratio of the area of the region \(R\) (transformed by the Jacobian) to the area of the rectangle \(S\) in the \(uv\)-plane. By comparing the areas computed in step 3, we get: \(Ratio = \frac{Area~of~R}{Area~of~S} \approx \frac{|J(u,v)| \Delta u \Delta v}{\Delta u \Delta v} = |J(u,v)|\).

Key concepts

Multivariable Calculus and the Jacobian

Multivariable calculus is the extension of Calculus to more than one variable. It includes the study of limits, derivatives, integrals, and summation of functions that depend on several variables. A key concept in multivariable calculus is the use of partial derivatives to understand how functions change when variables are varied independently.

One of the essential tools in multivariable calculus is the Jacobian matrix, which comprises all the first-order partial derivatives of a vector-valued function. It is significant because it can tell us how area or volume changes under a given transformation. For instance, when we transform an area in the uv-plane to a possibly distorted shape in the xy-plane, the Jacobian serves as a factor that relates the area of a small region in one plane to the area of the transformed region in the other plane. This is pivotal in applications such as physics and engineering where transformation of reference frames is commonplace.

Area Transformation Using the Jacobian

Area transformation is an important concept in multivariable calculus, especially when dealing with coordinate changes. The Jacobian determinant, usually represented as \(J(u, v)\), provides a measure of how much the area near the point \( (u,v) \) is magnified or reduced upon transformation to the xy-plane.

In the context of our problem, the Jacobian determinant is used to approximate the area of a parallelogram formed by transforming a rectangle in the uv-plane. The area of this parallelogram can be approximately calculated using the lengths of its sides, determined by the transformation functions \(g(u,v)\) and \(h(u,v)\), and then taking their cross product. For small changes, represented as \(\Delta u\) and \(\Delta v\), the transformation can be linearized, which simplifies the area calculation and vividly demonstrates the power of linear approximations in multivariable calculus.

Taylor Series Approximation in Transformations

The Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point. It is used to approximate functions by polynomials, where the degree of the polynomial determines the approximation's accuracy. In multivariable calculus, Taylor series can be used to approximate the behavior of functions of two or more variables near a specific point.

The exercise uses a Taylor series to approximate the functions \(g(u,v)\) and \(h(u,v)\) mapped from the uv-plane to the xy-plane. By considering a linear (first-order) approximation, which is a tangent line to the function, we can find how \(g\) and \(h\) change over small increments. This approximation is crucial when we calculate the transformed coordinates of points and find out how much the image of a rectangle from the uv-plane has been stretched or squashed in the xy-plane—the essence of understanding area transformation with multi-variable functions. When providing such explanations, elucidating the transition from the discrete points to their Taylor series approximations can significantly improve conceptual understanding.