Problem 47 - Advanced Engineering Mathematics

Question

Use the Chain Rule to find the indicated partial derivatives. $$ \begin{aligned} &w=\sqrt{x^{2}+y^{2}} ; x=\ln (r s+t u) \\ &y=\frac{t}{u} \cosh r s ; \frac{\partial w}{\partial t}, \frac{\partial w}{\partial r}, \frac{\partial w}{\partial u} \end{aligned} $$

Step-by-Step Solution

Answer

Compute partial derivatives of x and y, then use the chain rule for \(w\) with respect to each variable.

Step 1: Understand the Function Setup

The function is given as \(w = \sqrt{x^2 + y^2}\), where \(x\) and \(y\) are dependent on other variables \(r, s, t, u\). Specifically, \(x = \ln(r s + t u)\) and \(y = \frac{t}{u} \cosh(rs)\). We need to find the partial derivatives of \(w\) with respect to \(t\), \(r\), and \(u\).

Step 2: Compute Partial Derivatives of x and y

Find the partial derivatives of \(x\) and \(y\) with respect to \(t\), \(r\), and \(u\):- \(\frac{\partial x}{\partial t} = \frac{u}{rs + tu}\), \(\frac{\partial x}{\partial r} = \frac{s}{rs + tu}\), \(\frac{\partial x}{\partial u} = \frac{t}{rs + tu}\)- \(\frac{\partial y}{\partial t} = \frac{\cosh(rs)}{u}\), \(\frac{\partial y}{\partial r} = \frac{t}{u} s \sinh(rs)\), \(\frac{\partial y}{\partial u} = -\frac{t}{u^2} \cosh(rs)\).

Step 3: Use the Chain Rule for Partial Derivative of w with respect to t

Using the chain rule: \(\frac{\partial w}{\partial t} = \frac{x}{\sqrt{x^2 + y^2}} \frac{\partial x}{\partial t} + \frac{y}{\sqrt{x^2 + y^2}} \frac{\partial y}{\partial t}\).Substituting the derivatives:\[ \frac{\partial w}{\partial t} = \frac{x}{w} \cdot \frac{u}{rs+tu} + \frac{y}{w} \cdot \frac{\cosh(rs)}{u} \].

Step 4: Use the Chain Rule for Partial Derivative of w with respect to r

Using the chain rule: \(\frac{\partial w}{\partial r} = \frac{x}{\sqrt{x^2 + y^2}} \frac{\partial x}{\partial r} + \frac{y}{\sqrt{x^2 + y^2}} \frac{\partial y}{\partial r}\).Substituting the derivatives:\[ \frac{\partial w}{\partial r} = \frac{x}{w} \cdot \frac{s}{rs+tu} + \frac{y}{w} \cdot \left( \frac{t\sinh(rs)\cdot s}{u} \right) \].

Step 5: Use the Chain Rule for Partial Derivative of w with respect to u

Using the chain rule: \(\frac{\partial w}{\partial u} = \frac{x}{\sqrt{x^2 + y^2}} \frac{\partial x}{\partial u} + \frac{y}{\sqrt{x^2 + y^2}} \frac{\partial y}{\partial u}\).Substituting the derivatives:\[ \frac{\partial w}{\partial u} = \frac{x}{w} \cdot \frac{t}{rs+tu} + \frac{y}{w} \cdot \left( -\frac{t\cosh(rs)}{u^2} \right) \].

Step 6: Simplify Calculations

The expressions for the partial derivatives can be simplified further based on specific values of \(r, s, t, u\) or if needed, the expressions can be computed numerically for specific inputs where further simplification may not be straightforward algebraically.

Key Concepts

Partial Derivatives

Partial derivatives are derivatives of functions of multiple variables with respect to one variable, keeping other variables constant. In mathematical terms, when dealing with a function that depends on multiple variables, such as \( w = f(x, y) \), the partial derivative with respect to \( t \) focuses on how \( w \) changes as \( t \) changes, while \( r \), \( s \), and \( u \) are held constant.

A common notation for partial derivatives includes putting a \( \partial \) symbol in place of the \( d \) in typical derivatives. For example, the partial derivative of \( w \) with respect to \( t \) is written as \( \frac{\partial w}{\partial t} \).

In multivariable functions, these derivatives provide valuable insights:

• How sensitive the function is to changes in a particular variable.
• The independent effects of each variable on the function.

By understanding partial derivatives, students can analyze and identify the behavior of complex systems. This is especially useful in various fields such as physics, economics, and engineering.

Multivariable Calculus

Multivariable calculus explores the behaviors of functions with multiple independent variables. Unlike typical single-variable calculus, where functions depend on one variable, multivariable calculus lets you investigate how changing multiple factors influences a function.

In multivariable calculus, important concepts include:

• Functions of several variables: These describe systems where outcomes depend on more than one input, like \( w = \sqrt{x^2 + y^2} \) in the original problem.
• Partial derivatives: As previously discussed, these help analyze how functions change with each input variable independently.
• The Chain Rule: This relates to how composite functions change with respect to a change in one of their inputs.

When solving multivariable problems, visualizing functions and interpreting the topology of surfaces can become crucial. Graphical representations and using tools like contour plots or 3D graphs can help understand the relationships between variables.

Mathematical Functions

In mathematics, functions describe relationships between sets of inputs and outputs. A function typically has an independent variable whose alteration affects the value of the dependent variable.

Functions are versatile and can be defined for various types of variables and operations, often depicted as \( f(x) = y \).

• Composite functions: These combine two functions, where the output of one function becomes the input for another, as seen in \( w = \sqrt{x^2 + y^2} \), with \( x \) and \( y \) derived from other expressions.
• Parameterization: A technique to re-define relations among variables using different parameters, useful in multivariable contexts.
• Continuous functions: Where even small changes in the input cause small changes in the output, which is vital for differential calculus.

Understanding different types of mathematical functions and how they operate is fundamental for working with complex relationships in various scientific and engineering disciplines. It lays the groundwork for both analytical solutions and numerical simulations.