Problem 40 - Calculus Early Transcendentals

Question

Write the differential \(d w\) in terms of the differentials of the independent variables. $$w=f(x, y, z)=\sin (x+y-z)$$

Step-by-Step Solution

Answer

Question: Find the differential \(dw\) in terms of the differentials of independent variables \(dx\), \(dy\), and \(dz\) for the function \(w = f(x, y, z) = \sin(x + y - z)\). Answer: The differential \(dw\) is given by \(dw = \cos(x+y-z)dx + \cos(x+y-z)dy - \cos(x+y-z)dz\).

Step 1: Find the partial derivatives of f(x,y,z) with respect to x, y, and z

First, let's find the partial derivatives of the function \(f(x, y, z) = \sin(x + y - z)\) with respect to each independent variable: Partial derivative with respect to x: $$\frac{\partial{w}}{\partial{x}} = \frac{\partial}{\partial{x}}\sin(x+y-z) = \cos(x+y-z)$$ Partial derivative with respect to y: $$\frac{\partial{w}}{\partial{y}} = \frac{\partial}{\partial{y}}\sin(x+y-z) = \cos(x+y-z)$$ Partial derivative with respect to z: $$\frac{\partial{w}}{\partial{z}} = \frac{\partial}{\partial{z}}\sin(x+y-z) = -\cos(x+y-z)$$

Step 2: Find the differential dw

Now, we can use the partial derivatives to find the differential \(dw\): $$dw = \frac{\partial{w}}{\partial{x}}dx + \frac{\partial{w}}{\partial{y}}dy + \frac{\partial{w}}{\partial{z}}dz$$ Substitute the partial derivatives we found in Step 1: $$dw = \cos(x+y-z)dx + \cos(x+y-z)dy - \cos(x+y-z)dz$$ This is the differential \(dw\) in terms of the differentials of the independent variables \(dx\), \(dy\), and \(dz\).

Key Concepts

Partial Derivatives

In the realm of calculus, especially when dealing with functions of more than one variable, partial derivatives play a crucial role. When we have a multivariable function such as \( f(x, y, z) = \sin(x + y - z) \), we are interested in examining how this function changes as each of its variables changes independently.
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• Partial Derivative with Respect to \( x \): This is denoted as \( \frac{\partial w}{\partial x} \), and it involves differentiating \( \sin(x + y - z) \) with respect to \( x \) while treating \( y \) and \( z \) as constants. The result is \( \cos(x+y-z) \).
• Partial Derivative with Respect to \( y \): Similarly, \( \frac{\partial w}{\partial y} \) considers \( x \) and \( z \) as constants, yielding the same derivative of \( \cos(x+y-z) \).
• Partial Derivative with Respect to \( z \): Since \( z \) appears with a negative sign in the argument of the sine function, the partial derivative \( \frac{\partial w}{\partial z} \) becomes \(-\cos(x + y - z) \).

Using these derivatives, we can assess how the function varies with alterations in each independent variable, providing a glimpse into the function’s behavior with respect to each axis.

Differential Equations

Differential equations are equations that relate a function to its derivatives, offering insights into dynamic systems where change is continual. The differential \( dw \) is a linear approximation of the change in \( w \) as small changes occur in \( x, y, \) and \( z \):

• The equation \( dw = \frac{\partial w}{\partial x}dx + \frac{\partial w}{\partial y}dy + \frac{\partial w}{\partial z}dz \) makes use of each partial derivative found earlier.

What makes differential equations important is their ability to express the instantaneous rate of change in one quantity relative to changes in others.
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In our context:

• \( dx, dy, \) and \( dz \) are infinitesimally small changes in \( x, y, \) and \( z \) respectively.

By incorporating the individual contributions of each partial derivative, the differential \( dw \) captures the cumulative effect of all small changes in independent variables on \( w \), essentially expanding our understanding of dynamic relationships in multivariable scenarios.

Multivariable Calculus

In multivariable calculus, we extend the simple derivative concept from single-variable calculus to functions involving multiple variables. This extension allows us to investigate more complex relationships among various quantities. Our function \( w = \sin(x+y-z) \) is an example of a multivariable function, involving three independent variables.
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• Understanding Interactions: Multivariable calculus provides tools to evaluate how changes in one variable can influence others. It is akin to watching multiple streams merge into a river, where each contributes to the flow and behavior of the river.
• Applications: This branch is not limited to mathematical theory but is pivotal in fields like physics, engineering, and economics, where scenarios often depend on several varying factors.

These principles enable us to solve real-world problems by interpreting and predicting multidimensional changes, substantiated by deriving and understanding expressions like \( dw \) that give the total change through the differentials. Multivariable calculus, thus, equips us with the knowledge to tackle scenarios involving multiple factors efficiently.