Calculus

This appendix reviews elementary topics in single and multivariable calculus that are necessary to understand the theory of Euclidean tensor analysis developed in the main part of the notes.

Single variable calculus

We start with a brief review of single variable calculus. The primary object of study are functions of the form $f:I \subseteq \mathbb{R} \to \mathbb{R}$ that map an open interval $I = (a,b) \subseteq \mathbb{R}$ into the set of real numbers. Thus given any $x \in I$, $f(x)$ is a real number.

Continuity

Recall that $f:I \to \mathbb{R}$ is said to be continuous at $x \in I$ if, given any $\epsilon > 0$, there exists $\delta > 0$ such that $$|y - x| < \delta \quad\Rightarrow\quad |f(y) - f(x)| < \epsilon.$$ The idea is that as the point $y$ approaches $x$, the value of the function $f(y)$ approaches $f(x)$. In other words, there are no abrupt jumps in the function value at $x$. Note that the foregoing definition can be written equivalently as $$\lim_{y \to x} f(y) = f(x).$$ The function is said to be continuous over the whole of $I$ if it is continuous at every $x \in I$. The set of all continuous functions on $I$ is denoted as $C^0(I,\mathbb{R})$.

Differentiation

The function $f:I \to \mathbb{R}$ is said to be differentiable at $x \in I$ if the limit $$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$ exists. In this case, the is limit is called the derivative of $f$ at $x$ and is denoted variously as $$f'(x), \quad \dot{f}(x), \quad \frac{df(x)}{dx}, \quad Df(x),$$ depending on the context. The function $f$ is said to be differentiable on $I$ if it is differentiable at every $x \in I$. The set of all differentiable functions on $I$ whose derivative is also continuous on $I$ is denoted as $C^1(I,\mathbb{R})$.

Every differentiable function is continuous. To see this suppose that $f:I \to \mathbb{R}$ is a differentiable function. Then, at any $x \in I$, it can be seen that $$\lim_{y \to x} f(y) - f(x) = \lim_{y \to x} (y - x)\frac{f(y) - f(x)}{y - x} = 0 \cdot f'(x) = 0.$$ This shows that $\lim_{y \to x} f(y) = f(x)$, thereby establishing the claim. Note in particular that $C^1(I,\mathbb{R}) \subsetneq C^0(I,\mathbb{R})$.

Higher order derivatives can be defined similarly. For instance, a differentiable function $f:I \to \mathbb{R}$ is said to be twice differentiable at $x \in I$ if the function $f'(x):I \to \mathbb{R}$ is differentiable at $x$. In this case, the second derivative of $f$ at $x$ is written, variously, as $$f''(x), \quad \ddot{f}(x), \quad \frac{d^2f(x)}{dx^2}, \quad D^2f(x).$$ The function $f:I \to \mathbb{R}$ is said to be twice differentiable on $I$ if its second derivative exists at every $x \in I$. The set of all twice differentiable functions on $I$ whose second derivative is also continuous on $I$ is denoted as $C^2(I,\mathbb{R})$. Third and higher derivatives of $f$ are defined in an analogous manner. For instance, the set $C^k(I,\mathbb{R})$ consists of all functions $f:I \to \mathbb{R}$ that are $k$ times differentiable on $I$, and whose $k^{\text{th}}$ derivative is continuous on $I$. The function $f:I \to \mathbb{R}$ is said to be smooth on $I$ if it is an element of $C^k(I,\mathbb{R})$ for every $k \in \mathbb{N}$.

Properties of the derivative

We will now recall a few elementary properties of the derivative. The derivative is a linear operation in the following sense: given any differentiable functions $f:I \to \mathbb{R}$, $g:I \to \mathbb{R}$ and constants $a,b \in \mathbb{R}$, it follows from an elementary argument from the definition of the derivative that $$(a f + b g)'(x) = a f'(x) + b g'(x).$$ The product rule of differentiation states that $$(fg)'(x) = f'(x) g(x) + f(x) g'(x).$$ The product rule is also a simple consequence of the definition of the derivative.

An important property of the derivative pertains to composition of functions. For simplicity, let us consider differentiable functions $f, g: \mathbb{R} \to \mathbb{R}$. Then, the chain rule of differentiation states that $$(f \circ g)'(x) = f'(g(x)) g'(x).$$

Proof

To see this, note that $$\begin{split} (f \circ g)'(x) &= \lim_{h \to 0} \frac{f(g(x + h)) - f(g(x))}{h}\\ &= \lim_{h \to 0} \frac{f(g(x + h)) - f(g(x))}{g(x + h) - g(x)}\frac{g(x + h)) - g(x)}{h}. \end{split}$$ Noting that the differentiability of $g$ implies that $$\lim_{h \to 0} g(x + h) - g(x) = 0,$$ and introducing the notation $y = g(x + h) - g(x)$, we see that $$\begin{split} (f \circ g)'(x) &= \lim_{y \to 0} \frac{f(g(x) + y) - f(g(x))}{y}\lim_{h \to 0}{g(x + h) - g(x)}\frac{g(x + h)) - g(x)}{h}\\ &= f'(g(x))g'(x). \end{split}$$ This proves the chain rule of differentiation.

Example

As a simple application of the chain rule, consider the case of a bijective function $f:\mathbb{R} \to \mathbb{R}$ such that both $f$ and $f^{-1}$ are differentiable. The chain rule of differentiation can be used to related the derivatives of $f$ and $f^{-1}$ as follows: for any $x \in \mathbb{R}$, $$(f^{-1} \circ f)(x) = x \quad\Rightarrow\quad (f^{-1})'(f(x))f'(x) = 1 \quad\Rightarrow\quad (f^{-1})'(f(x)) = \frac{1}{f'(x)}.$$ A similar expression can be obtained by differentiating the identity $(f \circ f^{-1})(x) = x$.

Multivariable calculus

Let us now present certain elementary facts about multivariable calculus. To keep the discussion elementary, let us consider first the case of a real valued function of two real variables $f:\mathbb{R}^2 \to \mathbb{R}$. The development here parallels that presented in the single variable case. The general case of a function of many variables is considered subsequently.

Continuity

The function $f:R \subseteq \mathbb{R}^2 \to \mathbb{R}$, where $R$ is the open rectangle $(x_1,x_2) \times (y_1,y_2) \subseteq \mathbb{R}^2$, is said to be continuous at $(x,y) \in R$ if, for every $\epsilon > 0$, there exists a $\delta > 0$ such that $$\sqrt{(\bar{x} - x)^2 + (\bar{y} - y)^2} < \delta \quad\Rightarrow\quad |f(\bar{x},\bar{y}) - f(x,y)| \le \epsilon.$$ Notice how the Euclidean distance figures in this definition. As before, the definition of continuity can be equivalently restated as $$\lim_{(\bar{x},\bar{y}) \to (x,y)} f(\bar{x},\bar{y}) = f(x,y).$$ The intuitive explanation is the same as before: if $f$ is continuous at $(x,y) \in R$, then the value of $f$ close to $(x,y)$ is close to $f(x,y)$. In other words, there are no gaps in the range of $f$.

The function $f$ is said to be continuous on $R$ if it is continuous at every $(x,y) \in R$. In this case, the set of all continuous functions on $\mathbb{R}$ is written as $C^0(R,\mathbb{R})$.

Differentiability

Let us now study the differentiability of the function $f:R \to \mathbb{R}$. Rather than develop the general theory (this is presented in the main part of these notes), the presentation here provides an elementary account of the partial derivatives of $f$.

The function $f:R \to \mathbb{R}$ takes two real arguments and returns a real number. Fixing one of the arguments, we get a real-valued function of a real variable, which can be differentiated as studied before, in the context of single variable calculus. In the present case, fixing one of the arguments and letting the other vary results in two different limits: $$\lim_{h \to 0} \frac{f(x + h,y) - f(x,y)}{h}, \qquad \lim_{h \to 0} \frac{f(x,y + h) - f(x,y)}{h},$$ where $(x,y) \in R$ and $h \in \mathbb{R}$. When these limits exist, they are called the partial derivatives of $f$. Specifically, the partial derivatives of $f:R \to \mathbb{R}$ at $(x,y) \in \mathbb{R}$ are defined as $$\begin{split} \frac{\partial f(x,y)}{\partial x} &= \lim_{h \to 0} \frac{f(x + h,y) - f(x,y)}{h},\\ \frac{\partial f(x,y)}{\partial y} &= \lim_{h \to 0} \frac{f(x,y + h) - f(x,y)}{h}. \end{split}$$ The following alternative notations will be used for partial derivatives: $$\begin{split} \frac{\partial f(x,y)}{\partial x} &= \partial_x f(x,y) = \partial_1 f(x,y) = D_x f(x,y) = D_1 f(x,y),\\ \frac{\partial f(x,y)}{\partial y} &= \partial_y f(x,y) = \partial_2 f(x,y) = D_y f(x,y) = D_2 f(x,y). \end{split}$$

The foregoing discussion can be generalized to functions of the form $\mathsf{f}:\mathbb{R}^n \to \mathbb{R}^m$. Note that the function $\mathsf{f}$ can be written as a collection of $m$ functions $$\mathsf{f} = (f_1, \ldots, f_m),$$ where each $f_i:\mathbb{R}^n \to \mathbb{R}$, $i = 1, \ldots, m$ is a function that takes an $m$-tuple of real numbers and returns a real number. The partial derivaties of $\mathsf{f}$ at $\mathsf{x} = (x_1, \ldots, x_m) \in \mathbb{R}^m$can be arranged in the form of the $m \times n$ matrix $$\begin{bmatrix} \partial_1 f_1(\mathsf{x}) & \ldots & \partial_n f_1(\mathsf{x})\\ \vdots & \ddots & \vdots\\ \partial_1 f_m(\mathsf{x}) & \ldots & \partial_n f_m(\mathsf{x}) \end{bmatrix},$$ called the Jacobian of $\mathsf{f}$ at $\mathsf{x}$.

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Riemann integration

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