Function of Several Variables

Functions of several variables are mathematical constructs that take multiple input values and produce a single output value. Unlike functions of a single variable, which map real numbers to real numbers, functions of several variables map tuples of real numbers to real numbers.

Let’s consider a function $f(x_1,x_2,...,x_n)$, where $x_1,x_2,...,x_n$ are the input variables and $f$ is the output variable. Each input variable corresponds to a dimension in the domain space, and the output variable corresponds to the range space.

The general form of a function of several variables can be written as:

$f:{\mathbb{R}}^n\to \mathbb{R}$

Here, ${\mathbb{R}}^n$ represents the n-dimensional real space, and $\mathbb{R}$ represents the real numbers.

The function’s behavior can be visualized in multiple dimensions using graphs, contour plots, or level curves. For instance, a function of two variables, $f(x,y)$, can be visualized in three dimensions, where the $xy$-plane represents the input space, and the $z$-axis represents the output space.

Independent Variable

This is the input variable of a function. It is called “independent” because its value can be chosen freely without any constraints imposed by the function. In a function of several variables, there can be multiple independent variables. For example, in the function $f(x,y)$, both $x$ and $y$ can be independent variables.

Dependent Variable

This is the output variable of a function. It is called “dependent” because its value depends on the values of the independent variables and the function’s rule or relationship. The dependent variable is the one whose value is determined by the function’s behavior. In the function $f(x,y)$, $f$ is the dependent variable.

Domain

The domain of a function is the set of all possible input values (independent variables) for which the function is defined. It represents the valid inputs that the function can accept. For functions of several variables, the domain consists of all possible combinations of values for the independent variables that make the function meaningful. It is often denoted by $dom(f)$ or $D$.

Find the domain of the function f(x, y) = x^2 + y^2.

Since $f(x,y)$ is a polynomial function, it is defined for all real numbers. In other words, there are no restrictions on the values of $x$ and $y$. Therefore, the domain of $f(x,y)$ is all pairs of real numbers $(x,y)$.

In set notation, the domain can be expressed as:

$\text{Domain}=\{(x,y)\mid x,y\in \mathbb{R}\}$

This indicates that $x$ and $y$ can take any real number as their values without any restrictions.

Let's consider the function g(x,y) = \frac{1}{x-y}.

To find the domain of this function, we need to consider the values of $x$ and $y$ for which the denominator $x-y$ is not equal to zero, because division by zero is undefined.

So, we set $x-y\ne 0$ and solve for $x$ and $y$:

$x-y\ne 0$

This implies that $x$ cannot be equal to $y$. In other words, $x$ and $y$ cannot take the same value simultaneously.

herefore, the domain of $g(x,y)$ is all pairs of real numbers $(x,y)$ where $x\ne y$.

In set notation, the domain can be expressed as:

$\text{Domain}=\{(x,y)\mid x\ne y,\text{ where }x,y\in \mathbb{R}\}$

This indicates that $x$ and $y$ can be any real numbers as long as they are not equal to each other.

Let's consider the function h(x,y) = \sqrt{x^2-y}.

To find the domain of this function, we need to ensure that the expression under the square root, $x^2-y$, is non-negative because taking the square root of a negative number is undefined in the real number system.

So, we set $x^2-y\ge 0$ and solve for $x$ and $y$:

$x^2-y\ge 0$

This inequality implies that $x^2\ge y$, which means that $y$ must be less than or equal to $x^2$ for the function to be defined.

Therefore, the domain of $h(x,y)$ is all pairs of real numbers $(x,y)$ where $y\le x^2$.

In set notation, the domain can be expressed as:

$\text{Domain}=\{(x,y)\mid y\le x^2,\text{ where }x,y\in \mathbb{R}\}$

This indicates that $x$ and $y$ can be any real numbers as long as $y$ is less than or equal to the square of $x$.

Range

The range of a function is the set of all possible output values (dependent variables) that the function can produce. It represents the set of values that the function can take on as its output. For functions of several variables, the range is the set of all possible values of the dependent variable that can result from varying the independent variables over the domain. It is often denoted by $range(f)$ or $R$.

Example

To find the range of the function $f(x,y)=x^2+y^2$, we need to consider all possible values that the function can attain for different choices of $x$ and $y$.

Since both $x^2$ and $y^2$ are always non-negative (as squares of real numbers), their sum $x^2+y^2$ will also be non-negative.

Furthermore, any non-negative real number can be obtained as $x^2+y^2$ for some choices of $x$ and $y$. For example, if we set $x=0$ and $y=0$, then $x^2+y^2=0$. Moreover, if we let $x$ and $y$ vary over all real numbers, we can obtain any non-negative real number as $x^2+y^2$.

Therefore, the range of the function $f(x,y)=x^2+y^2$ is all non-negative real numbers, which can be expressed as:

$\text{Range}=\{f(x,y)\mid f(x,y)\ge 0,\text{ where }x,y\in \mathbb{R}\}$

In interval notation, the range can be represented as $[0,+\infty )$, indicating that the range includes all non-negative real numbers starting from 0 up to positive infinity.

Example

To find the range of the function $g(x,y)=\frac{1}{x-y}$, we need to consider all possible values that the function can attain for different choices of $x$ and $y$.

However, before we proceed, it’s important to note that the function $g(x,y)$ is not defined for all possible values of $x$ and $y$ since division by zero is not allowed. Specifically, the function is undefined when $x=y$ because the denominator $x-y$ becomes zero.

Thus, we need to exclude these cases from our consideration. For the remaining values of $x$ and $y$ where $x\ne y$, the function $g(x,y)$ will take real values.

Therefore, the range of the function $g(x,y)=\frac{1}{x-y}$ can be expressed as:

$\text{Range}=\left\{g(x,y)|x\ne y,\text{ where }x,y\in \mathbb{R}\right\}$

In other words, the range consists of all real numbers except when the denominator $x-y$ becomes zero, which occurs when $x=y$.

Example

To find the range of the function $h(x,y)=\sqrt{x^2-y}$, we need to consider all possible values that the function can attain for different choices of $x$ and $y$.

However, before we proceed, it’s important to note that the expression under the square root, $x^2-y$, must be non-negative for the function to be defined in the real number system. This means that $x^2\ge y$.

Thus, the range of $h(x,y)$ includes all real numbers that are non-negative square roots of numbers of the form $x^2-y$, where $x^2\ge y$.

Therefore, the range of the function $h(x,y)=\sqrt{x^2-y}$ can be expressed as:

$\text{Range}=\left\{h(x,y)|x^2\ge y,\text{ where }x,y\in \mathbb{R}\right\}$

In other words, the range consists of all real numbers that are non-negative square roots of numbers of the form $x^2-y$, where $x^2\ge y$.

Limits

The limit of a function $f(x,y)$ as $(x,y)$ approaches a point $(a,b)$ is denoted as: ${\lim }_{(x,y)\to (a,b)}f(x,y)=L$

Approach

Similar to single-variable limits, to find the limit of a function of two variables, we examine the behavior of the function as the input $(x,y)$ approaches the point $(a,b)$.

Notation

• $(x,y)$ represents the independent variables.
• $(a,b)$ represents the point towards which the variables are approaching.
• $L$ represents the limit value.

Evaluation

The limit exists if the function approaches the same value $L$ regardless of the direction from which $(x,y)$ approaches $(a,b)$.

Formal Definition

The limit of $f(x,y)$ as $(x,y)$ approaches $(a,b)$ is $L$ if for every $ϵ>0$, there exists a $\delta >0$ such that if $0<\sqrt{(x-a{)}^2+(y-b{)}^2}<\delta$, then $|f(x,y)-L|<ϵ$.

Example

Consider the function $f(x,y)=\frac{x^{2}y}{x^4+y^2}$ and the point $(0,0)$. To find the limit as $(x,y)$ approaches $(0,0)$, we need to evaluate the function as $(x,y)$ gets arbitrarily close to $(0,0)$.

Conclusion

Limits for functions of two variables extend the concept from single-variable functions and are essential for understanding continuity, differentiability, and the behavior of functions in multivariable calculus.

Continuity

In calculus, continuity is a fundamental concept that describes the behavior of a function at every point in its domain. When extending this concept to functions of two variables, we consider how the function behaves as both variables change simultaneously.

Definition

A function $f(x,y)$ is continuous at a point $(a,b)$ if the following conditions are met:

1. $f(a,b)$ is defined (i.e., the function exists at $(a,b)$).
2. The limit of $f(x,y)$ as $(x,y)$ approaches $(a,b)$ exists.
3. The limit of $f(x,y)$ as $(x,y)$ approaches $(a,b)$ is equal to $f(a,b)$.

Visual Interpretation

In visual terms, continuity for functions of two variables means that the surface defined by the function has no breaks, jumps, or holes at the point $(a,b)$. This implies that as we move along the surface towards $(a,b)$, the function values approach $f(a,b)$ smoothly from all directions.

Examples

• A polynomial function of two variables is continuous everywhere.
• Rational functions are continuous where they are defined, excluding points where the denominator is zero.
• Piecewise-defined functions may or may not be continuous, depending on the behavior of each piece at their joining points.

Importance

Continuity is essential in analyzing the behavior of functions and understanding their properties. It allows us to make predictions about the function’s behavior and facilitates the application of calculus techniques such as differentiation and integration.

Conclusion

Understanding continuity for functions of two variables is crucial in various fields such as engineering, physics, economics, and computer science. It provides insights into the behavior of multivariable functions and enables the application of mathematical tools to solve real-world problems.