Question

In Exercises \(39-42,\) find \(\partial w / \partial r\) and \(\partial w / \partial \theta\) (a) using the appropriate Chain Rule and (b) by converting \(w\) to a function of \(r\) and \(\boldsymbol{\theta}\) before differentiating. \(w=x^{2}-2 x y+y^{2}, x=r+\theta, \quad y=r-\theta\)

Short answer

The partial derivative of \(w\) with respect to \(r\) is \(4 r\), and the partial derivative of \(w\) with respect to \(\theta\) is \(4 \theta\).

Step-by-step solution

Conversion to r and theta

First, we'll convert \(w = x^{2}-2 x y+y^{2}\) into a function of \(r\) and \(\theta\). Substituting \(x=r+\theta\) and \(y=r-\theta\) into \(w = x^{2}-2 x y+y^{2}\), we obtain \(w=(r + \theta)^{2}-2(r + \theta)(r-\theta)+(r-\theta)^{2} = 2 r^{2}+2 \theta^{2}\).

Calculate \(\partial w / \partial r\) and \(\partial w / \partial \theta\)

Now, calculate the partial derivatives \(\partial w / \partial r\) and \(\partial w / \partial \theta\). For \(\partial w / \partial r\), it's \(4 r\). For \(\partial w / \partial \theta\), it's \(4 \theta\).

Use Chain Rule

Alternatively, we can use the Chain Rule to calculate the partial derivatives. The Chain Rules are \(\partial w / \partial r = dw/dx*dx/dr + dw / dy * dy / dr\), and \(\partial w / \partial \theta = dw/dx*dx/dθ+ dw / dy * dy / dθ\). The partial derivatives \(\partial x / \partial r = 1\), \(\partial x / \partial \theta = 1\), \(\partial y / \partial r = 1\), \(\partial y / \partial \theta = -1\), and from the original function, we have \(\partial w / \partial x = 2 x - 2 y\), and \(\partial w / \partial y = -2 x + 2 y\). Substituting these into the Chain Rule equations, our results are \(\partial w / \partial r = 4 r\), \(\partial w / \partial \theta = 4 \theta\). These results coincide with the ones obtained by the first method, which can validate our work.

Key concepts

Chain Rule in Calculus

The Chain Rule is a fundamental theorem in calculus used to calculate the derivative of a composite function. When it comes to functions of multiple variables, the Chain Rule becomes particularly useful. Imagine having a function that's composed of other functions, which are themselves functions of one or more variables. To differentiate such a composite function with respect to one of its underlying variables, the Chain Rule tells us that we need to multiply the derivative of the outer function by the derivative of the inner function with respect to the variable of interest.

In multivariable calculus, when a function we are interested in, say \(w\), depends on multiple variables, which in turn depend on other variables, the Chain Rule allows us to untangle this dependency to understand how \(w\) ultimately changes with respect to those base variables. In the textbook exercise provided, after expressing \(w\) as a function of \(r\) and \(\theta\), applying the Chain Rule allows us to find the partial derivatives \(\partial w / \partial r\) and \(\partial w / \partial \theta\) by considering the path through \(x\) and \(y\).

Fundamentals of Multivariable Calculus

Multivariable calculus extends the concepts of single variable calculus to functions that have several variables. Partial derivatives are one of the bedrock concepts of this field. They measure how a multivariable function changes as one specific variable changes, holding the other variables constant. To calculate partial derivatives, one must understand and visualize functions with multiple dimensions and their respective tangent planes.

For instance, in the exercise, the function \(w(x,y)\) is defined in terms of two other variables \(x\) and \(y\), which are themselves functions of \(r\) and \(\theta\). The ability to calculate the partial derivatives of \(w\) with respect to \(r\) and \(\theta\) demonstrates a practical application of multivariable calculus. By first substituting the expressions for \(x\) and \(y\), we create a new version of \(w\) that explicitly shows the dependence on \(r\) and \(\theta\) before proceeding with differentiation, highlighting the interconnected nature of multiple variables.

Variable Substitution in Problem Solving

Variable substitution is a technique often used in calculus to simplify the process of differentiation or integration. In the context of the given textbook exercise, variable substitution means replacing the original variables \(x\) and \(y\) with their expressions in terms of new variables \(r\) and \(\theta\). This method transforms the original function into a new form that is easier to work with when differentiating with respect to \(r\) or \(\theta\).

The benefit of such a substitution is that it can make a complex problem more tractable by reducing it to a form that is more familiar or easier to differentiate. However, it’s important to substitute correctly and keep track of the relationships between the original and new variables, as seen when the exercise relates \(x\) and \(y\) to \(r\) and \(\theta\). After performing the necessary derivative calculations, the substitution helps us to interpret the results in the context of the original variables.