10.1. Calculus on Euclidean Space

10.1.1. Euclidean Space

Definition

Euclidean 3-space \(\mathbf{R}^3\) is the set of all ordered triples of real numbers. Such a triple \(\mathbf{p} = (p_1, p_2, p_3)\) is called a point of \(\mathbf{R}^3\).

  • \(\mathbf{R}^3\) is a vector space over the real numbers.

    \[\begin{split}&\mathbf{p} = (p_1, p_2, p_3)\\ &\mathbf{q} = (q_1, q_2, q_3)\\ &\mathbf{p}+\mathbf{q} = (p_1 + q_1, p_2 +q_2, p_3 + q_3)\\ &a\mathbf{p} = (ap_1, ap_2, ap_3)\\ &\mathbf{0} = (0,0,0)\end{split}\]

Definition

Let \(x,y,z\) be the real-valued functions on \(\mathbf{R}^3\) such that for each point \(\mathbf{p} = (p_1, p_2, p_3)\)

\[\begin{split}x(\mathbf{p}) = p_1\\ y(\mathbf{p}) = p_2\\ z(\mathbf{p}) = p_3\\\end{split}\]

These functions \(x,y,z\) are called the natural coordinate functions of \(\mathbf{R}^3\).

We also write:

\[x_1 = x, \quad x_2 = y, \quad x_3 = z\]

We thus have:

\[\mathbf{p} = (p_1, p_2, p_3) = (x_1(\mathbf{p}), x_2(\mathbf{p}), x_3(\mathbf{p}))\]

Definition

A real-valued function \(f\) on \(\mathbf{R}^3\) is differentiable (or infinitely differentiable or smooth) provided all partial derivatives of \(f\), of all orders, exist and are continuous.

  • \((f+g)(\mathbf{p}) = f(\mathbf{p}) + g(\mathbf{p})\)

  • \((fg)(\mathbf{p}) = f(\mathbf{p})g(\mathbf{p})\)

  • In the following all functions are assumed to be differentiable real valued functions unless stated otherwise.

Differentiation

  • Is always a local operation.

  • The domain of \(f\) need not be \(\mathbf{R}^3\). It needs to be only an open set of \(\mathbf{R}^3\).