Problem 37

Question

Express the following in $\partial$ notation. (a) $f_{y y y}$ (b) $f_{x x y}$ (c) $f_{x y y y}$

Step-by-Step Solution

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Answer

(a) $\frac{\partial^3 f}{\partial y^3}$; (b) $\frac{\partial^3 f}{\partial x^2 \partial y}$; (c) $\frac{\partial^4 f}{\partial x \partial y^3}$.

1Step 1: Understand Partial Notation

Partial notation abla denotes taking the derivative with respect to variables independently while holding others constant. It utilizes the symbol $ \partial $. For example, the partial derivative of a function $f$ with respect to $x$ is written as $ \frac{\partial f}{\partial x} $.

2Step 2: Express $f_{y y y}$ in Partial Notation

The expression $f_{y y y}$ indicates taking the third derivative of function $f$ with respect to $y$ three times. In partial notation, this is written as: \[ \frac{\partial^3 f}{\partial y^3} \]

3Step 3: Express $f_{x x y}$ in Partial Notation

The expression $f_{x x y}$ means taking the derivative first with respect to $x$ twice, and then once with respect to $y$. In partial notation, this is: \[ \frac{\partial^3 f}{\partial x^2 \partial y} \]

4Step 4: Express $f_{x y y y}$ in Partial Notation

The expression $f_{x y y y}$ indicates taking the derivative first with respect to $x$ once, and then with respect to $y$ three times. In partial notation, this is written as: \[ \frac{\partial^4 f}{\partial x \partial y^3} \]

Key Concepts

Partial NotationHigher-Order DerivativesCalculus Notation

Partial Notation

In calculus, partial notation is essential when dealing with functions of multiple variables. It allows us to differentiate with respect to one variable while treating others as constants. This is particularly helpful when examining the behavior of functions where variables influence each other, such as in economics or physics. The partial derivative is denoted by the symbol $ \partial $, distinguishing it from the standard derivative symbol $ d $.

To express a partial derivative, we use the format $ \frac{\partial f}{\partial x} $ to represent the derivative of the function $ f $ with respect to the variable $ x $.
If we want to take the derivative with respect to $ y $ once again after already differentiating by $ x $, it is expressed as $ \frac{\partial^2 f}{\partial x \partial y} $.

These notations help indicate not only which variable is differentiated but also the order in which these operations occur. It is crucial to follow the proper sequence because the outcome can vary depending on how differentiation is applied.

Higher-Order Derivatives

Higher-order derivatives involve taking the derivative of a function multiple times. In the context of partial derivatives, this means differentiating more than once with respect to one or more variables. Higher-order derivatives provide insights into the curvature and rate of change of functions, enabling deeper analysis.

A second-order derivative with respect to $ x $ is shown as $ \frac{\partial^2 f}{\partial x^2} $, and it helps us understand how the slope of $ f $ changes.
For mixed derivatives, like differentiating once with $ x $ and once with $ y $, it appears as $ \frac{\partial^2 f}{\partial x \partial y} $.
Expressions such as $ \frac{\partial^3 f}{\partial y^3} $ denote the third derivative with respect to $ y $, shedding light on more complex changes in the function's behavior.

These derivatives are especially useful in fields that require modeling multidimensional processes, like statistical regression or in the study of differential equations.

Calculus Notation

Calculus notation is the language for expressing mathematical concepts in calculus succinctly and clearly. It includes various symbols to convey operations, changes, and relationships effectively. Understanding these notations is essential for anyone studying calculus.

The basic notation for derivatives in single-variable calculus uses $ \frac{d}{dx} $, while multi-variable calculus employs $ \frac{\partial}{\partial x} $ to indicate partial derivatives.
When dealing with higher-order derivatives, a superscript indicates the number of times the derivative is taken, such as in $ \frac{\partial^3}{\partial x^2 \partial y} $.
Function notation often accompanies derivatives, shaping expressions like $ f'(x) $ or $ f''(x) $ for first and second derivatives in single-variable calculus.

By mastering calculus notation, students can efficiently communicate their ideas and follow complex operations in mathematical problem-solving. It's a toolkit that is invaluable in both theoretical analysis and practical applications of calculus.

Problem 37

Problem 38

Other exercises in this chapter

Problem 37

Describe geometrically the level surfaces for the functions defined in Problems 33-38. $$ f(x, y, z)=4 x^{2}-9 y^{2} $$

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Problem 37

Let $f(x, y)=x^{2} y /\left(x^{4}+y^{2}\right)$. (a) Show that $f(x, y) \rightarrow 0$ as $(x, y) \rightarrow(0,0)$ along any straight line $y=m x$. (b)

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Problem 38

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}}

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Problem 39

Suppose that the temperature $T$ on the circular plate $\left\\{(x, y): x^{2}+y^{2} \leq 1\right)$ is given by $T=2 x^{2}+y^{2}-y_{\text {. Find }}$ the h

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