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Chapter 11: Problem 67

Define the total differential of a function of two variables.

Short Answer

Expert verified
The total differential of a function of two variables \(f(x, y)\) is the sum of the partial differentials of that function with respect to its variables, each multiplied by the differential of the variable itself. In mathematical notation, this can be represented as \(df = \frac{鈭俧}{鈭倄} * dx + \frac{鈭俧}{鈭倅} * dy\).

Step by step solution

01

Definition

The total differential of a function of two variables \(f(x, y)\) is defined by the following formula: \[df(x, y) = \frac{鈭俧}{鈭倄} * dx + \frac{鈭俧}{鈭倅} * dy\]
02

Calculation of partial derivatives

To compute the total differential, first calculate the partial derivatives of the function \(f\) with respect to each variable \(x\) and \(y\). Those are denoted \(\frac{鈭俧}{鈭倄}\) and \(\frac{鈭俧}{鈭倅}\).
03

Substitution of differential terms

Substitute the computed partial derivatives back into the definition of the total differential. If possible, simplify the resulting expression to a more manageable form. The resulting expression represents the total differential, or the infinitesimal change in the function for infinitesimal changes in the variables \(x\) and \(y\).

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Most popular questions from this chapter

Consider the function \(w=f(x, y),\) where \(x=r \cos \theta\) and \(y=r \sin \theta .\) Prove each of the following. (a) \(\frac{\partial w}{\partial x}=\frac{\partial w}{\partial r} \cos \theta-\frac{\partial w}{\partial \theta} \frac{\sin \theta}{r}\) \(\frac{\partial w}{\partial y}=\frac{\partial w}{\partial r} \sin \theta+\frac{\partial w}{\partial \theta} \frac{\cos \theta}{r}\) (b) \(\left(\frac{\partial w}{\partial x}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^{2}=\left(\frac{\partial w}{\partial r}\right)^{2}+\left(\frac{1}{r^{2}}\right)\left(\frac{\partial w}{\partial \theta}\right)^{2}\)

Describe the change in accuracy of \(d z\) as an approximation of \(\Delta z\) as \(\Delta x\) and \(\Delta y\) increase.

Describe the difference between the explicit form of a function of two variables \(x\) and \(y\) and the implicit form. Give an example of each.

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. \(w=\arctan \frac{y}{x}, \quad x=r \cos \theta, \quad y=r \sin \theta\)

In Exercises 33 and \(34,\) find \(d^{2} w / d t^{2}\) using the appropriate Chain Rule. Evaluate \(d^{2} w / d t^{2}\) at the given value of \(t\) \(w=\arctan (2 x y), \quad x=\cos t, \quad y=\sin t, \quad t=0\)
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