0.1 Important notions

Functions. $\mathbb{R}^n\to \mathbb{R}$

send points to numbers (scalars)

Curves. $I \to \mathbb{R}^n$

send numbers to points (in space)

Vector fields. $\mathbb{R}^n \to \mathbb{R}^n$

sends points to vectors (in the same dimension)

1 Baby linear algebra and geometry

1.1 Babby’s first vectors

Read $\operatorname{comp}_{\mathbf{a}} \mathbf{b}, \operatorname{proj}_{\mathbf{a}} \mathbf{b}$ leftwards as component/projection of $\mathbf{b}$ on $\mathbf{a}$. $$\begin{align*} \operatorname{comp}_{\mathbf{a}} \mathbf{b} &= \frac{\mathbf{a}\cdot\mathbf{b}}{\left\lVert \mathbf{a} \right\rVert} \\ \operatorname{proj}_{\mathbf{a}} \mathbf{b} &= \frac{\mathbf{a}\cdot\mathbf{b}}{\mathbf{a}\cdot\mathbf{a}} \\ \mathbf{a}\cdot\mathbf{b} &= \left\lVert \mathbf{a} \right\rVert\left\lVert \mathbf{b} \right\rVert \cos \theta \\ \left\lVert \mathbf{a} \right\rVert^2 &= \mathbf{a}\cdot\mathbf{a} \\ \left\lVert \mathbf{a}\times\mathbf{b} \right\rVert &= \left\lVert \mathbf{a} \right\rVert\left\lVert \mathbf{b} \right\rVert \sin\theta \end{align*}$$

• Use $\times$ to find area of parallelogram or shortest distance to line from point (divide by $\left\lVert \mathbf{a} \right\rVert$).
• When doing cross product remember to minus the $\mathbf{j}$ term!
• The $\cdot$ and $\times$ operators are bilinear.
• Triple product = determinant of $3$ vectors.

1.2 (Translated) subspaces

Any hyperspace can be described by a normal vector $\mathbf{n}$. Equation of any hyperspace can be given by $\mathbf{r}\cdot\mathbf{n} = 0$, translation gives $\mathbf{n}\cdot\mathbf{r} = \mathbf{n}\cdot\mathbf{r}_0$ where $\mathbf{r}_0$ is the position vector of any given point in the hyperspace.

• Angle between hyperspaces = angle between their normals.
• Shortest distance between $Q$ and $\Pi$ = $\operatorname{comp}_{\mathbf{n}} \overrightarrow{AQ}$ on $\mathbf{n}$ where $A\in \Pi$.

1.3 Vector functions

• Differentiate vector $\mathbf{r}(t)$ component-wise.
• Product rule generalises to $\cdot$ and $\times$, proof by bloody expansion.
• Arc length formula. For arc traced by $\mathbf{r}(t)$, from $a \leq t \leq b$, arc length is $\int_a^b \left\lVert \mathbf{r}'(t) \right\rVert \ dt$.

2 Surfaces & Multivariable functions

Cylinder.

Exists plane $\Pi$ such that all planes parallel to $\Pi$ intersect $S$ in the same curve. (Common cylinders are those with a free variable.)

If all else fails, fix $1$ variable and trace the other two.

3 Limits & Continuity

• To show limit does not exist at $\mathbf{a}$, trace along several different paths (any line that passes through $\mathbf{a}$ will do) and get a contradiction (not well-defined).
• To show limit exist, use limit laws or squeeze. 
  On limit laws: for $\left\{\, \text{functions of }\mathbf{x}\text{ with limit at }\mathbf{a} \,\right\}$,
  • $\lim_{\mathbf{x}\to\mathbf{a}}$ is linear.
  • $\lim_{\mathbf{x}\to\mathbf{a}}$ respects $\cdot$ and $/$ provided $\lim$ denominator is not $0$.
• $f$ is continuous at $\mathbf{a}$ iff substitution property $$\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = f(\mathbf{a})$$
• Subspace of continuous functions is closed under $\pm,\cdot,/$ and $\circ$.
• Standard continuous functions are polynomials, trig, exp and rational functions.

4 Differential Calculus

4.1 Partial derivatives

Suppose $f: \mathbb{R}^n \to \mathbb{R}$, let $(e_i)_{i=1,2,\dots,n}$ be standard basis for $\mathbb{R}^n$. $$D_i f = \frac{\partial f}{\partial e_i} =_{\text{def}} \lim_{h\to 0} \frac{f(\mathbf{v} + he_i) - f(\mathbf{v})}{h}.$$ When differentiating normally is too hard use $\uparrow$.

Clairaut’s Theorem. Partial differentiation operators commute. $\forall i,j.~ D_i D_j = D_j D_i$.

4.2 Differentiability and linear approximation

$f: \mathbb{R}^n \to \mathbb{R}$ is differentiable at $\mathbf{a}$ iff

• all the partial derivatives $D_1 f,\dots,D_n f$ (these terms are in fact limits) exist at the point $\mathbf{a}$, and
• there exists a function $\varepsilon: \mathbb{R}^n \to \mathbb{R}^n$ (defined for small $\mathbf{h}$ such that) $$\lim_{\mathbf{h}\to\mathbf{0}} \varepsilon(\mathbf{h}) = \mathbf{0},\quad\textbf{and}$$ $$f(\mathbf{a} + \mathbf{h}) - f(\mathbf{a}) = \nabla f(\mathbf{a})\cdot \mathbf{h} + \varepsilon(\mathbf{h})\cdot \mathbf{h}$$ post-composing $\varepsilon$ with tuple-selection map will give $n$ functions, and
• in $2$ variables where $\mathbf{h} = (\Delta x, \Delta y)$, it simplifies to Stewart’s shit-tier definition $$\Delta z = f_x \Delta x + f_y \Delta y + \varepsilon_1(\mathbf{h}) \Delta x + \varepsilon_2(\mathbf{h}) \Delta y$$

On differentiability, only these implications hold, at a point $\mathbf{x}\in \mathbb{R}^n$,

1. all partial derivatives continuous at $\mathbf{x}$ $\implies$ $f$ differentiable at $\mathbf{x}$,
2. $f$ differentiable $\implies$ $f$ continuous,
3. $f$ differentiable $\implies$ all partial derivatives exist.

Suppose $f$ is differentiable at $\mathbf{a}$, then for a small vector $\mathbf{h}$, $$f(\mathbf{a} + \mathbf{h}) \approx f(\mathbf{a}) + \nabla f(\mathbf{a}) \cdot \mathbf{h}$$

4.3 Chain rule

$f$ is a differentiable function, $\mathbf{r}(t)$ is a differentiable curve. Then $f(\mathbf{r}(t))$ is differentiable and $$\frac{df(\mathbf{r}(t))}{dt} = \nabla f(\mathbf{r}(t))\cdot \mathbf{r}'(t),$$ or in terms of coordinates, if $\mathbf{r}(t) = \left\langle r_1(t),\dots,r_n(t) \right\rangle$, $$\frac{df(\mathbf{r}(t))}{dt} = \frac{\partial f}{\partial r_1}\frac{d r_1}{d t} + \dots + \frac{\partial f}{\partial r_n}\frac{d r_n}{d t}.$$

In general, suppose $f(\mathbf{x}(\mathbf{t}))$ is differentiable and $\mathbf{x}(\mathbf{t}) = \left\langle x_1(\mathbf{t}),\dots,x_n(\mathbf{t}) \right\rangle$, then $$\frac{\partial f}{\partial t_i} = \frac{\partial f}{\partial x_1}\frac{\partial x_1}{\partial t_i} + \dots + \frac{\partial f}{\partial x_n}\frac{\partial x_n}{\partial t_i}.$$ (remember as adding all paths traced from $f$ to $t_i$.)

Corollary.

$\nabla f =_{\text{def}} \left\langle D_1f , \dots , D_nf \right\rangle$ is always normal to the level curve/surface of $f$.

Implicit differentiation.

Suppose $F(x,y,z) = 0$ implicitly defines $z = \phi(x)$, then by chain rule (assuming $y$ independent) $$\frac{\partial F}{\partial x} + \frac{\partial F}{\partial z}\frac{\partial z}{\partial x} = 0.$$

Tangent Plane shortcut.

For $f: \mathbb{R}^2\to \mathbb{R}$, vector normal to tangent plane at $(a,b)$ is given by $$\left\langle 1,0,f_x(a,b) \right\rangle \times \left\langle 0,1,f_y(a,b) \right\rangle = \left\langle -f_x(a,b),-f_y(a,b), 1 \right\rangle.$$

4.4 Directional derivatives

Given a unit vector $\mathbf{u}$, the derivative of $f$ in the direction of $\mathbf{u}$ is $$D_{\mathbf{u}} f =_\text{def} \nabla f \cdot \mathbf{u}.$$

Min/max rate of change of $f$ at a point $P$,

• max rate of change is in direction $\nabla f$, with magnitude $\left\lVert \nabla f(P) \right\rVert$, conversely
• min rate of change is in direction $-\nabla f$, with magnitude $-\left\lVert \nabla f(P) \right\rVert$

4.5 Extrema

(Local min/max was not defined rigorously so I assume no1cur.)

Critical point.

$P$ is a critical point of $f$ if $\nabla f(P) = \mathbf{0}$.

Boundary point.

$P$ is a boundary point of $S$ if every open ball centered at $P$ contains a point in $S$ and a point not in $S$.

Closed set.

$S$ is closed if it contains all boundary points.

Bounded.

There exists $b$ such that for any point $X$ in set, we have $\left\lVert X \right\rVert \leq b$.

Extreme Value Theorem. For a function $f$ defined on a closed and bounded set $S$, $f$ has global maximum and minimum in $S$.

Closed Interval Method for $f:D\to \mathbb{R}$, where $D$ is closed and bounded.

1. Find values of $f$ at all critical points in $D$.
2. Trace values of $f$ at each boundary of $D$.
3. min/max

4.5.1 Lagrange Multipliers

Suppose $f,g$ are differentiable functions where $\nabla g \ne \mathbf{0}$. Optimisation: find min/max $f$ (objective) given $g(x,y) = k$ (constraint).

1. solve $\nabla f(x,y) = \lambda \nabla g(x,y)$, where $\lambda$ is Lagrange Multiplier to be determined.
2. Evaluate $f$ at all points found, proceed to min/max.

Higher-dimensional cases are analogous.

5 Integral Calculus

5.1 Additive property of integrals

Let $\mathcal{F} = \left\{\, R_i \,\right\}_{i\in I}$ be a finite family of sets (regions, curves, surfaces) such that $R_i\cap R_j$ is of measure $0$ whenever $i\ne j$, then $$\int_{\mathcal{F}} = \sum_{i\in I} \int_{R_i}$$ (obvious from Riemann sum definition).

5.2 Double Integrals

If domain of $f$ can be represented as $D = \left\{\, (x,y): a\leq x\leq b, h_1(x)\leq y\leq h_2(x) \,\right\}$, then $$\iint_D f(x,y)\ dA = \int_a^b \int_{h_1(x)}^{h_2(x)} f(x,y)\ dy\ dx.$$

5.2.1 Polar Regions

If the polar region of integral is given as $D = \left\{\, (r,\theta): 0\leq a\leq r\leq b, h_1(r)\leq \theta\leq h_2(r) \,\right\}$ (angle depends on radius), then $$\iint_D f(r,\theta)\ dA = \int_a^b \int_{h_1(r)}^{h_2(r)} f(r, \theta)r \ d\theta\ dr.$$ On the other hand when radius depends on angle, say $D = \left\{\, (r,\theta): \alpha\leq\theta\leq\beta, h_1(\theta)\leq r\leq h_2(\theta) \,\right\}$, $$\iint_D f(r,\theta)\ dA = \int_\alpha^\beta \int_{h_1(\theta)}^{h_2(\theta)} f(r, \theta)r \ dr\ d\theta.$$

5.3 Triple Integrals

Synopsis. Reduce the triple integral into a double integral by projecting the domain of integration on a coordinate plane.

Suppose $D\in\mathbb{R}^2$ such that $E = \left\{\, (x,y,z): (x,y)\in D, h_1(x,y) \leq z \leq h_2(x,y) \,\right\}$, then an integral over $E$ can be reduced to a double integral by doing $$\iiint_E f(x,y,z)\ dV = \iint_D\left[ \int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z)\ dz\right]\ dA.$$

5.3.1 Cylindrical coordinates

Nothing much, use $\uparrow$ to reduce into a double integral over polar region.

5.3.2 Spherical coordinates

$$(\rho,\theta,\phi)\mapsto (\rho\sin\phi\cos\theta,\rho\sin\phi\sin\theta,\rho\cos\phi)$$

1. This shit simplifies computations involving spheres (no shit) and cones.
2. the Jacobian is $$dx\ dy\ dz = \rho^2\sin\phi\ d\rho\ d\theta\ d\phi$$

6 Interlude on Linear Algebra

Suppose $T: (u,v)\mapsto (x,y)$, then $$\iint_{T(R)} f(x,y)\ dx\ dy = \iint_{R} (f\circ T)(u,v)\left\lvert \det(T') \right\rvert\ du\ dv$$ where $T' = \displaystyle\frac{\partial (x,y)}{\partial (u,v)}$. (I need a commutative diagram for this shit.)

Find $R$ by tracing $T^{-1}$ along boundary regions of $T(R)$. Higher-dimensional cases are analogous.

7 Vector Calculus

7.1 Potential functions

Conservative.

$\mathbf{F}$ is a conservative vector field if there exists $\varphi: \mathbb{R}^n\to\mathbb{R}$ such that $\mathbf{F} = \nabla \varphi$.

Test for conservativeness.

Whenever $\mathbf{F}$ is defined in an open and simply-connected (no holes) region. Suppose $\mathbf{F} = \left\langle D_1\varphi,D_2\varphi,\dots,D_n\varphi \right\rangle$ for some $\varphi$, then test if all 2nd-order partial derivatives of $\varphi$ indeed commute.

In short, $\nabla\times\mathbf{F} = \mathbf{0}$.

Find potential function when $\mathbf{F}$ is defined on box.

If eye power fails, do partial integrations on $\mathbf{F}$.

Fundamental Theorem for Curve Integrals.

If there exists $\varphi: \mathbb{R}^n\to\mathbb{R}$ such that $\mathbf{F} = \nabla \varphi$, let $P,Q$ be start and end of curve $C$ respectively, then $$\int_{P,C}^Q \mathbf{F}\cdot d\mathbf{r} = \varphi(Q) - \varphi(P).$$

Corollaries.

1. When $\mathbf{F}$ admits a potential function, integral of $F$ is independent of the path $C$ joining $P$ and $Q$.
2. If $C$ is a closed path (same start/end point) and $\int_C \mathbf{F} \ne 0$, then $\mathbf{F}$ is not conservative. (contrapositive)

7.2 Line & Surface integrals

Notation. Let everything be as differentiable as needed.

1. $f$ denotes a scalar field.
2. $\mathbf{F}$ denotes a vector field.
3. $C$ denotes a smooth ($C^1$) curve.
   • Let $\mathbf{r}(t): I\to\mathbb{R}^n$ be a parametrisation.
4. $S$ denotes a surface.
   • Let $\mathbf{r}(u,v): D \to\mathbb{R}^3$ be a parametrisation.
   • Then $\mathbf{n} = \pm \displaystyle\frac{\mathbf{r}_u \times \mathbf{r}_v}{\left\lVert \mathbf{r}_u \times \mathbf{r}_v \right\rVert}$ is a unit normal vector

Formulas will be $$\begin{gather*} \int_C f \ d\mathbf{r} = \int_I f(\mathbf{r}(t)) \left\lVert \mathbf{r}'(t) \right\rVert\ dt \tag*{(1)} \\ \int_C \mathbf{F}\cdot d\mathbf{r} = \int_I \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\ dt \tag*{(2)} \\ \iint_S f \ dS = \iint_D f(\mathbf{r}(u,v)) \left\lVert \mathbf{r}_u \times \mathbf{r}_v \right\rVert\ dA \tag*{(3)} \\ \iint_S \mathbf{F}\cdot \ d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u,v))\cdot \left(\mathbf{r}_u \times \mathbf{r}_v\right) \ dA \tag*{(4)} \end{gather*}$$

Notes.

2. Computationally we write (example in $2$-space) as symbolic shorthand $$\int_C \mathbf{F}\cdot d\mathbf{r} = \int_I f_1\ dx + f_2\ dy.$$

3. Computations can be skipped if $\mathbf{F}$ is conservative, refer to section above.

4. Be careful of orientation conventions for $\mathbf{n}$, possibly switching the cross product.

   When $\mathbf{r}(x,y) = \left\langle x,y,g\left(x,y\right) \right\rangle$, $\mathbf{r}_x \times \mathbf{r}_y = \left\langle -g_x, -g_y, 1 \right\rangle$. Computationally rewrite (4) as $$\iint_S \mathbf{F}\cdot \ d\mathbf{S} = \pm \iint_D -f_1 g_x - f_2 g_y + f_3 \ dy\ dx.$$

7.3 Divergence and curl

Divergence.

$$\nabla\cdot \mathbf{F} = \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} + \frac{\partial f_3}{\partial z}$$

Curl.

$$\nabla\times \mathbf{F} = \begin{vmatrix} \mathbf{i}&\mathbf{j}&\mathbf{k}\\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y}&\frac{\partial }{\partial z}\\ f_1&f_2&f_3 \end{vmatrix}$$

For any vector field $\mathbf{F}$, $\nabla\cdot\left(\nabla\times\mathbf{F}\right) = 0$

7.4 Theorems that let you interchange stuff

Green’s Theorem (2d).

Let $C$ be a closed path and $A$ be region bounded by $C$, $\mathbf{F} = \left\langle f_1,f_2 \right\rangle$, $$\oint_C \mathbf{F} = \oint_C f_1\ dx + f_2\ dy = \iint_A \left(\frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y}\right) \ dy\ dx.$$ Convention: $A$ lies to the left of the curve $C$.

Gauss’ Divergence Theorem.

Let $S$ be boundary surface of $E$ with outward orientation, then $$\iint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_E \nabla\cdot\mathbf{F} \ dV.$$

Stokes’ Theorem.

Let $C$ be boundary curve of surface $S$ with unit normal $\mathbf{n}$. Let $C$ be positively oriented ($\mathbf{r}'(t)\times \mathbf{n}$ points out), then $$\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_S \nabla\times\mathbf{F}\cdot d\mathbf{S}.$$

8 Trig identities

$$\begin{pmatrix} \cos(\alpha+\beta) &-\sin(\alpha+\beta) \\ \sin(\alpha+\beta) & \cos(\alpha+\beta) \end{pmatrix} = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix} \begin{pmatrix} \cos\beta & -\sin\beta \\ \sin\beta & \cos\beta \end{pmatrix}$$

8.1 Pre-computing spherical stuff

Fix $\rho$, let $\mathbf{r}$ parametrise something, $$\begin{align*} \mathbf{r}(\phi,\theta) &= \rho\begin{pmatrix} \sin\phi\cos\theta\\ \sin\phi\sin\theta\\ \cos\phi \end{pmatrix}; & \mathbf{r}_{\phi} &= \rho\begin{pmatrix} \cos\phi\cos\theta \\ \cos\phi\sin\theta \\ -\sin\phi \end{pmatrix} & \mathbf{r}_{\theta} &= \rho\begin{pmatrix} -\sin\phi\sin\theta \\ \sin\phi\cos\theta \\ 0 \end{pmatrix} \end{align*}$$ then $\mathbf{r}_{\phi}\times \mathbf{r}_{\theta}$ points outwards, and $$\begin{align*} \mathbf{r}_{\phi}\times \mathbf{r}_{\theta} &= \rho^2 \begin{pmatrix} -\sin^2\phi \cos\theta \\ \sin^2\phi\sin\theta \\ \cos\phi\sin\phi \end{pmatrix} \end{align*}$$