Problem 101 - Calculus 3

Question

At what points in space is \(g(x, y, z)=x^{2}+y^{2}-2 z^{2}\) continuous?

Step-by-Step Solution

Answer

The function is continuous everywhere in \(\mathbb{R}^3\).

Step 1: Identify the Type of Function

The function given is \(g(x, y, z) = x^2 + y^2 - 2z^2\). Notice that this is a polynomial function of three variables: \(x\), \(y\), and \(z\).

Step 2: Apply the Continuity Property of Polynomial Functions

In calculus, it is known that polynomial functions are continuous everywhere in their domain. Since polynomials are defined across all of \(\mathbb{R}^3\), this function is continuous for all \(x\), \(y\), and \(z\) values.

Step 3: Conclusion of the Continuity

Since polynomial functions are continuous everywhere in their domain, and this function is defined for all real numbers, \(g(x, y, z)\) is continuous at every point in space.

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers, combined using addition, subtraction, and multiplication. An example of a polynomial function is given by \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \] where \(a_n, a_{n-1}, \ldots, a_0\) are coefficients and \(n\) is a non-negative integer.
Polynomials have several important properties:

• They are defined and continuous for all real numbers.
• The degree of a polynomial gives information about its broad shape and possible roots.
• Simple operations like addition, subtraction, and multiplication on polynomials result in another polynomial.

With multivariable polynomials, like the one in our problem, each variable is part of the polynomial and can independently take any real value. This makes them particularly nice for ensuring continuity in different spaces.

Multivariable Calculus

In multivariable calculus, we expand our understanding from functions of one variable to functions of several variables. Consider a function that depends on variables \(x, y, z\): \[ g(x, y, z) = x^2 + y^2 - 2z^2 \]This is a polynomial function in three variables. Multivariable functions map points in higher-dimensional space (such as the three-dimensional space \(\mathbb{R}^3\)) to a single real number.
Key concepts in multivariable calculus include:

• Understanding how changes in any variable affect the function output.
• The ability to compute derivatives with respect to each variable, known as partial derivatives.
• The importance of understanding the gradient, which points in the direction of the greatest rate of increase of the function.

When evaluating continuity in multivariable calculus, we are interested in whether small changes in our input produce small changes in the output across all variables simultaneously.

Continuous Functions

Continuity is a fundamental concept in calculus that describes how smoothly a function behaves. A function \(f(x)\) is said to be continuous at a point \(x = a\) if the following are true:- The function \(f(x)\) is defined at \(x = a\).- The limit of \(f(x)\) as \(x\) approaches \(a\) exists.- The limit of \(f(x)\) as \(x\) approaches \(a\) is equal to \(f(a)\).
For multivariable functions, like \(g(x, y, z) = x^2 + y^2 - 2z^2\), the function is continuous if it is continuous in each variable when all others are held constant. In simpler terms, if you can swap any variable and see a smooth transition without sudden jumps or breaks, it's continuous.
Since polynomial functions, by their nature, do not have points of discontinuity, they are continuous everywhere. This is because:

• There are no undefined points in their domain.
• Limits can easily be evaluated and match the function values.

Continuous functions are predictable and do not have any sudden breaks, which makes them important in both theoretical and practical applications.