Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the following in subscript notation. (a) \(\frac{\partial^{3} f}{\partial x^{2} \partial y}\) (b) \(\frac{\partial^{4} f}{\partial x^{2} \partial y^{2}}\) (c) \(\frac{\partial^{5} f}{\partial x^{3} \partial y^{2}}\)

Step-by-Step Solution

Answer

(a) \( f_{xxy} \); (b) \( f_{xxyy} \); (c) \( f_{xxxyy} \)

Step 1: Understanding Subscript Notation

Subscript notation is an alternative way to express partial derivatives of a function. In this notation, the derivative orders are represented as subscripts next to the function, denoting which variable has been differentiated and how many times.

Step 2: Transforming (a) to Subscript Notation

The expression \( \frac{\partial^{3} f}{\partial x^{2} \partial y} \) indicates the third partial derivative of \( f \), where \( x \) is differentiated twice and \( y \) once. This can be written in subscript notation as \( f_{xxy} \).

Step 3: Transforming (b) to Subscript Notation

For \( \frac{\partial^{4} f}{\partial x^{2} \partial y^{2}} \), the fourth partial derivative of \( f \) involves differentiating with respect to \( x \) and \( y \) each twice. Hence, it is expressed as \( f_{xxyy} \) in subscript notation.

Step 4: Transforming (c) to Subscript Notation

The expression \( \frac{\partial^{5} f}{\partial x^{3} \partial y^{2}} \) specifies that \( f \) is differentiated with respect to \( x \) three times and \( y \) two times. This is written in subscript notation as \( f_{xxxyy} \).

Key Concepts

Subscript Notation

When working with partial derivatives, it's often useful to use subscript notation. This alternative method provides a concise way to express the order and the variables involved in differentiation.

• In subscript notation, the variable and the number of times the function is differentiated with respect to that variable are represented as subscripts next to the function.
• For example, if you have the partial derivative \( \frac{\partial^{3} f}{\partial x^{2} \partial y} \), it means that the function \( f \) has been differentiated twice with respect to \( x \) and once with respect to \( y \).
• This can be compactly written as \( f_{xxy} \).

This notation can be particularly helpful when dealing with more complex or higher order derivatives, as it simplifies expressions and makes them easier to manage.

Multivariable Calculus

Multivariable calculus is the mathematical framework for analyzing functions of multiple variables. It extends the concepts of calculus to higher dimensions.

• While single-variable calculus deals with functions of one variable, multivariable calculus focuses on functions with more than one independent variable.
• Partial derivatives are a core aspect of this discipline, allowing us to explore how a function changes with respect to one variable while holding other variables constant.
• These partial derivatives help in exploring the behavior and nature of multidimensional spaces.

Applications of multivariable calculus are vast, ranging from optimizing complex systems to developing models in physics and engineering.

Higher Order Derivatives

Higher order derivatives provide insights into the deeper behavior of functions beyond standard first derivatives. These derivatives are imperative in applications where the rate of change itself changes.

• The first derivative of a function gives us the rate of change or slope of the function.
• By taking derivatives again, we obtain higher order derivatives, which reveal information about the curvature or concavity of the function.
• In a multivariable context, second and higher order partial derivatives allow for analysis of interactions between multiple variables.

To express these in mathematical terms, higher order derivatives use combinations of partial derivatives with respect to different variables, allowing for a more comprehensive understanding of complex functions.