# Functions of Several Variables

• kzang
In summary, to find the equations of circles, if any, where the given sphere intersects each coordinate plane, we set one coordinate equal to 0 and solve for the remaining variables. For the points of intersection, we set two coordinates equal to 0 and solve for the remaining variable.

## Homework Statement

a.) Find the equations of the circles (if any) where the sphere (x-1)^2+(y+3)^2+(z-2)^2=4 intersects each coordinate plane.
b.)Find the points (if any) where this sphere intersects each coordinate axis.

## The Attempt at a Solution

a.) The sphere intersects the y-z plane so x=0. So the equation would be (y+3)^2+(z-2)^2=4?

(x-1)^2+(y+3)^2+(z-2)^2=4

if x = 0, you get
(-1)^2+(y+3)^2+(z-2)^2=4

for b) on the z aaxis, y=x=0

For a) you should put x=0. Yes. But you didn't put x=0, you put x-1=0. For b) the x-axis is defined by y=0 and z=0.

## 1. What is a function of several variables?

A function of several variables is a mathematical relationship between multiple independent variables and one dependent variable. It takes in multiple input values and produces a single output value.

## 2. How is a function of several variables different from a function of one variable?

A function of one variable has only one independent variable, while a function of several variables has multiple independent variables. This means that the output of a function of several variables can be affected by changes in multiple input variables, rather than just one.

## 3. What is the domain of a function of several variables?

The domain of a function of several variables is the set of all possible input values for the independent variables. It is important to consider the domain when working with functions of several variables, as certain inputs may not be valid or produce a meaningful output.

## 4. How are partial derivatives used in functions of several variables?

Partial derivatives are used to find the rate of change of a function with respect to each independent variable. This allows us to determine the direction and rate of change in each input variable, which can be useful for optimization and understanding the behavior of the function.

## 5. What are some real-world applications of functions of several variables?

Functions of several variables are used in a variety of fields, such as economics, physics, and engineering. They can be used to model complex systems and relationships between multiple variables, and can help in making predictions and optimizing outcomes.