Functions of Several Variables
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Functions of Several Variables
A (real) function of \(n\) variables is a function \(f:\mathbb{R}^n \to \mathbb{R}\). It’s domain is a subset of \(\mathbb{R}^n\), and we can think about it as a way to assign each point in the domain a real number.
Examples of domains
Find the domain of the function \(f(x,y,z) = \ln(36 - x^2 - y^2 - z^2)\):
Here we must have \(36 - x^2 - y^2 - z^2 > 0\), or \(x^2 + y^2 + z^2 < 36\) which describes the interior of the sphere with center at the origin and with radius 6. Another name for this is an open ball with center at the origin and radius 6.
Find the domain of the function \(g(x,y) = \sqrt{y^2 - x^2}\):
Here we must have \(y^2 - x^2 \ge 0\), or \(y^2 \ge x^2\). Since both sides of the inequality are positive and square root is an increasing function, we can take square root of both sides and get \(\left\lvert y\right\rvert \ge \left\lvert x\right\rvert\) which describes the part of the plane at and above the graph of \(y = \left\lvert x\right\rvert\) or at and below the graph of \(y = -\left\lvert x\right\rvert\).
Find the domain of the function \(h(x,y,z) = \sqrt{z^2 - x^2 - y^2}\):
Here we must have \(z^2 - x^2 - y^2 \ge 0\), or \(z^2 \ge x^2 + y^2\). Since both sides of the inequality are positive and square root is an increasing function, we can take square root of both sides and get \(\left\lvert y\right\rvert \ge \sqrt{x^2 + y^2}\) which describes the part of the space at and inside the cone \(y = \pm\sqrt{x^2 + y^2}\).