Problem 23 - Calculus Early Transcendentals

Question

Solve the following relations for \(x\) and \(y,\) and compute the Jacobian \(J(u, v)\) $$u=x+y, v=2 x-y$$

Step-by-Step Solution

Answer

Answer: -1/3

Step 1: Solve the given relations for x and y

Add the two given relations to eliminate the variable y: $$(u=x+y)+(v=2x-y)=u+v=3x$$ Now, we can easily find x in terms of u and v: $$x=\frac{u+v}{3}$$ Next, plug the x value found in either of the relations to find y: $$u=x+y \Rightarrow y=u-x$$ Plug the x value we found earlier: $$y=u-\frac{u+v}{3}$$ So we have found x and y in terms of u and v: $$x = \frac{u+v}{3}$$ $$y = u-\frac{u+v}{3}$$

Step 2: Compute the partial derivatives

Find the partial derivatives of x and y with respect to u and v: $$\frac{\partial x}{\partial u} = \frac{\partial}{\partial u} \frac{u+v}{3} = \frac{1}{3}$$ $$\frac{\partial x}{\partial v} = \frac{\partial}{\partial v} \frac{u+v}{3} = \frac{1}{3}$$ $$\frac{\partial y}{\partial u} = \frac{\partial}{\partial u} \left(u-\frac{u+v}{3}\right) = 1 - \frac{1}{3} = \frac{2}{3}$$ $$\frac{\partial y}{\partial v} = \frac{\partial}{\partial v} \left(u-\frac{u+v}{3}\right) = - \frac{1}{3}$$

Step 3: Compute the Jacobian matrix and determinant

Form the Jacobian matrix J(u, v) using the partial derivatives we found in step 2: $$J(u, v)= \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix} = \begin{pmatrix} \frac{1}{3} & \frac{1}{3}\\ \frac{2}{3} & -\frac{1}{3} \end{pmatrix}$$ Now, compute the determinant of the Jacobian matrix: $$|J(u, v)| = \frac{1}{3} \cdot (-\frac{1}{3}) - \frac{1}{3} \cdot \frac{2}{3} = -\frac{1}{9} - \frac{2}{9} = -\frac{3}{9} = -\frac{1}{3}$$ Thus, the Jacobian J(u, v) is equal to \(-\frac{1}{3}\).

Key Concepts

Partial Derivatives

Understanding partial derivatives is crucial when we study functions that depend on multiple variables. In essence, a partial derivative measures how a function changes as one of its variables is varied, with all other variables held constant. This concept extends the idea of a derivative from single-variable calculus to the richer field of multivariable calculus.

For example, if we have a function like temperature over a geographical area, where temperature depends on both latitude and longitude, a partial derivative with respect to latitude would reveal how the temperature changes if you move north or south, assuming longitude remains fixed.

To compute a partial derivative, we use the same differentiation rules as in single-variable calculus, but apply them to one variable at a time. In the exercise provided, we computed the partial derivatives of our functions with respect to each variable, enabling us to understand how each of the new variables, u and v, affects the original variables x and y independently.

Multivariable Calculus

The study of multivariable calculus expands upon the foundational concepts encountered in single-variable calculus, such as limits, derivatives, and integrals, to functions of several variables. It's essential for describing and analyzing physical phenomena where multiple factors are at play, like in fluid dynamics, electromagnetism, or financial models.

In our context, solving for relations between variables, as done in the provided exercise, is an example of the application of multivariable calculus. Here, we worked with functions of two variables, but the concepts can extend to any number of variables. This branch of calculus is also where we encounter concepts such as gradient, divergence, curl, and, pertinent to our exercise, the Jacobian determinant which captures how changes in input variables lead to changes in output variables in a multi-dimensional setting.

Jacobian Matrix

The Jacobian matrix is a powerful tool in multivariable calculus used to study how multivariable functions behave. It is a matrix of all first-order partial derivatives of a vector-valued function. In more practical terms, it is like a multi-dimensional version of the derivative you meet in single-variable calculus, telling us about the change of each function as we move in each direction of the space defined by the variables.

In the exercise, after calculating the partial derivatives of the functions relative to each of the new variables u and v, these values were used to construct the Jacobian matrix. The determinant of this matrix, known as the Jacobian determinant, gives significant insight; it tells us about the