Problem 21 - Calculus with Concepts in Calculus

Question

Determine \(d f\). $$ f(x, y, z)=x e^{y^{2}-z^{2}} $$

Step-by-Step Solution

Answer

The differential is \(df = e^{y^2 - z^2} dx + 2xy e^{y^2 - z^2} dy - 2xz e^{y^2 - z^2} dz\).

Step 1: Understand the Problem

We are given a function \(f(x, y, z) = x e^{y^2 - z^2}\) and need to find its differential \(df\). This involves finding the partial derivatives of the function with respect to each variable and then applying the formula for the total differential.

Step 2: Find Partial Derivatives

First, we find the partial derivative of \(f\) with respect to \(x\): \[\frac{\partial f}{\partial x} = e^{y^2 - z^2}\] Then, the partial derivative with respect to \(y\): \[\frac{\partial f}{\partial y} = x \cdot e^{y^2 - z^2} \cdot 2y = 2xy e^{y^2 - z^2}\] Finally, the partial derivative with respect to \(z\): \[\frac{\partial f}{\partial z} = x \cdot e^{y^2 - z^2} \cdot (-2z) = -2xz e^{y^2 - z^2}\]

Step 3: Write the Differential

Using the formula for the total differential: \[ df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz\] Substituting the partial derivatives: \[ df = e^{y^2 - z^2} dx + 2xy e^{y^2 - z^2} dy - 2xz e^{y^2 - z^2} dz\] Thus, \(df\) is fully expressed in terms of \(dx\), \(dy\), and \(dz\).

Key Concepts

Partial Derivatives

In multivariable calculus, a partial derivative is a derivative where you fix all but one of the variables. This means you differentiate with respect to one variable while treating the others as constants.
This concept is especially important when dealing with functions of multiple variables, such as in the given function, \(f(x, y, z) = x e^{y^2 - z^2}\).
We have three variables: \(x\), \(y\), and \(z\). The partial derivative of \(f\) with respect to each variable gives the rate at which \(f\) changes as that particular variable changes, while the others remain constant.

• Partial Derivative with respect to \(x\): Here, \(y\) and \(z\) are constants. Differentiation yields \(\frac{\partial f}{\partial x} = e^{y^2 - z^2}\).
• Partial Derivative with respect to \(y\): Now, \(x\) and \(z\) are constants, and we find \(\frac{\partial f}{\partial y} = 2xy e^{y^2 - z^2}\).
• Partial Derivative with respect to \(z\): Here, \(x\) and \(y\) are constants, resulting in \(\frac{\partial f}{\partial z} = -2xz e^{y^2 - z^2}\).

Understanding partial derivatives is crucial for exploring how each input variable influences the function's output.

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus and involves more than one variable. It is particularly useful for dealing with functions that depend on multiple inputs.
In the function \(f(x, y, z) = x e^{y^2 - z^2}\), we are considering how each of the three input variables affects the output. Each variable can influence the outcome differently, and multivariable calculus provides the tools to analyze these effects precisely.
In contrast to single-variable calculus, where we find derivatives with respect to one variable, in multivariable calculus, we use gradients and partial derivatives to understand changes in multiple directions. This helps in applications like physics, engineering, and more, where systems often depend on various simultaneous factors.

• Total Differential: This is a significant tool in multivariable calculus. It approximates the value of the function based on small changes in the input variables, using partial derivatives.
• Geometrical Interpretation: Functions of multiple variables can often be visualized as surfaces in three-dimensional space. The total differential helps to approximate slopes or gradients of these surfaces, contributing to understanding their geometry.

Through multivariable calculus, we get a comprehensive look at how each part of a system interacts with every other part.

Differential Calculus

Differential calculus is the branch of calculus revolving around the concept of a derivative, which represents how a function changes as its inputs change.
For multivariable functions, like \(f(x, y, z) = x e^{y^2 - z^2}\), this means understanding how changes in each variable affect the function's output. The total differential \(df\) encapsulates these changes in consolidated form.

• Calculating the Total Differential: Once you have partial derivatives, they are combined to form the total differential. For instance, \(df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz\).
• Practical Application: This is useful for estimating how outputs change in response to small changes in any or all of the variable inputs. It gives insight into the function's sensitivity and response to varying conditions.

By carefully studying differential calculus, particularly in a multivariable context, you gain a powerful understanding of intricate relationships in mathematical models.