Parametrized Curve:

$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$

$$\frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{dy/dt}{dx/dt} \right) = \frac{d}{dt} \left( \frac{d}{dx} \left( \frac{dy/dt}{dx/dt} \right) \right) = \frac{d}{dt} \left( \frac{d/dt (dy/dt)}{dx/dt} \right)$$

$$\text{Area} = \int_{\alpha}^{\beta} y(t) x'(t) , dt$$

$$\text{Arc Length} = \int_{\alpha}^{\beta} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} , dt$$

Polar Coordinates:

$$x = r \cos \theta, \quad y = r \sin \theta, \quad r = \sqrt{x^2 + y^2}, \quad \tan \theta = \frac{y}{x}$$

$$\text{Area} = \frac{1}{2} \int_{\alpha}^{\beta} r^2 , d\theta$$

$$\text{Arc Length} = \int_{\alpha}^{\beta} \sqrt{r^2 + \left( \frac{dr}{d\theta} \right)^2} , d\theta$$

Vectors:

$$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}| \cos \theta$$

$$|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}| \sin \theta$$

$$\mathbf{a} \cdot \mathbf{b} = 0 \implies \text{orthogonal}$$

$$\mathbf{r}(t) = \mathbf{r_0} + t\mathbf{v}$$

$$\mathbf{v} = \mathbf{r_1} - \mathbf{r_0}$$

$$\text{proj}_{\mathbf{a}} \mathbf{b} = \mathbf{a} \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|^2}$$

$$\text{comp}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|}$$

Derivatives:

$$a(x-x_0) + b(y-y_0) + c(z-z_0) = 0$$

$$D = \frac{|a x_1 + b y_1 + c z_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$$

$$\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \quad \mathbf{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle \quad \int_a^b \mathbf{r}(t) , dt = \left\langle \int_a^b f(t) , dt, \int_a^b g(t) , dt, \int_a^b h(t) , dt \right\rangle$$

$$\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}$$

Tangent Plane:

$$z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$$

Linearization:

$$L(x,y) - f(a,b) \approx f_x(a,b)(x - a) + f_y(a,b)(y - b)$$

Chain Rule:

$$\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}$$

If $F(x,y) = 0$:

$$\frac{dy}{dx} = -\frac{F_x}{F_y}, \quad \frac{\partial z}{\partial x} = -\frac{F_x}{F_z}, \quad \frac{\partial z}{\partial y} = -\frac{F_y}{F_z}$$

Gradient:

$$\nabla f = \langle f_x, f_y \rangle$$

$$D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u} = |\nabla f| |\mathbf{u}| \cos \theta = |\nabla f| \cos \theta$$

Tangent Plane to Level Surface:

$$F_x(x_0, y_0, z_0)(x - x_0) + F_y(x_0, y_0, z_0)(y - y_0) + F_z(x_0, y_0, z_0)(z - z_0) = 0$$

Critical Points:

$$f_x(a,b) = f_y(a,b) = 0$$

$$D(a,b) = f_{xx} f_{yy} - (f_{xy})^2$$

(If $D > 0$ and $f_{xx} > 0$: local minimum) (If $D > 0$ and $f_{xx} < 0$: local maximum) (If $D < 0$: saddle point) (If $D = 0$: inconclusive)

Level Sets: $f(x,y)=k\ \text{(curve) or } F(x,y,z)=k\ \text{(surface)}$

Direction of level curves: directional derivative is $0$, which is perpendicular to the gradient.

$\nabla f$ is orthogonal to the level curve $f=k$ and points in the direction of the maximum rate of increase $|\nabla f|$. $\nabla F(x,y,z)$ is the normal vector to the tangent plane, thus, the level set $F=k$ at $P(x_0,y_0,z_0)$

Continuity (Smooth functions like polynomials/exp/trig are continuous wherever defined): $f$ is continuous at $(a,b)$ if $f(a,b)$ is defined, $\lim_{x,y \to a,b} f(x,y)$ exists, and $\lim f(x,y) = f(a,b)$

Differentiability (Equivalent to existence of tangent plane): $f$ is differentiable at $(a,b)$ if

$$f(a+h, b+k) = f(a,b) + f_x(a,b)h + f_y(a,b)k + o(\sqrt{h^2 + k^2})$$

Lagrange Multipliers:

One Constraint: Solve

$$\nabla f = \lambda \nabla g$$

and $g(x, y, z) = k$.

Multiple Constraints: Solve

$$\nabla f=\lambda\nabla g + \mu\nabla h$$

$$g(x,y,z)=k_1$$

$$h(x,y,z)=k_2$$

Triple Integrals:

General:

$$\iiint_{E} f(x,y,z),dV$$

Type I:

$$\iiint_{E} f(x,y,z),dV = \iint_{D} \left[ \int_{z=u_1(x,y)}^{z=u_2(x,y)} f(x,y,z), dz \right] dA$$

Type II:

$$\iiint_{E} f(x,y,z),dV = \iint_{D} \left[ \int_{x=u_1(y,z)}^{x=u_2(y,z)} f(x,y,z), dx \right] dA$$

Type III:

$$\iiint_{E} f(x,y,z),dV = \iint_{D} \left[ \int_{y=u_1(x,z)}^{y=u_2(x,z)} f(x,y,z), dy \right] dA$$

Polar/Cylindrical Coordinates:

$$\iint_R f(x, y), dy, dx = \iint_D f(r \cos \theta, r \sin \theta), r, dr, d\theta$$

$$x = r \cos \theta \qquad y = r \sin \theta \qquad z = z \qquad x^2 + y^2 = r^2$$

$$dV = r, dz, dr, d\theta$$

Spherical Coordinates:

$$x = \rho \sin \phi \cos \theta \qquad y = \rho \sin \phi \sin \theta \qquad z = \rho \cos \phi$$

$$x^2 + y^2 + z^2 = \rho^2$$

$$dx, dy, dz = \rho^2 \sin \phi, d\rho, d\phi, d\theta$$

Center of Mass (Double/Triple Integral):

$$M = \iint_D \rho(x, y), dA$$

$$\bar{x} = \frac{1}{M} \iint_D x \rho(x, y), dA \qquad \bar{y} = \frac{1}{M} \iint_D y \rho(x, y), dA$$

($M_y = \iint_D x \rho(x, y), dA \Rightarrow \bar{x} = \frac{M_y}{M}$, $M_x = \iint_D y \rho(x, y), dA \Rightarrow \bar{y} = \frac{M_x}{M}$)

Change of Variable (Jacobian):

$$J = \frac{\partial(x, y)}{\partial(u, v)} = \begin{vmatrix} x_u & x_v \ y_u & y_v \end{vmatrix}$$

$$\iint_R f(x, y), dx, dy = \iint_S f(x(u, v), y(u, v)) |J|, du, dv$$

Line Integral with respect to Arc Length ($ds$):

$$\int_{C}f(x,y,z),ds=\int_{a}^{b}f(x(t),y(t),z(t))\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}+\left(\frac{dz}{dt}\right)^{2}},dt$$

$$\int_{C}f(x,y),ds=\int_{a}^{b}f(x(t),y(t))\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}},dt$$

Line Integral of a Scalar Function (with respect to coordinates):

$$\int_{C}P,dx+Q,dy+R,dz=\int_{a}^{b}\left(P\frac{dx}{dt}+Q\frac{dy}{dt}+R\frac{dz}{dt}\right)dt$$

$$\int_{C}f(x,y),dx=\int_{a}^{b}f(x(t),y(t))\cdot x'(t),dt$$

$$\int_{C}f(x,y),dy=\int_{a}^{b}f(x(t),y(t))\cdot y'(t),dt$$

Line Integral of a Vector Field (Work Integral):

$$\int_{C}\mathbf{F}\cdot d\mathbf{r}=\int_{a}^{b}\mathbf{F}(\mathbf{r}(t))\cdot \mathbf{r}^{\prime}(t),dt=\int_{a}^{b}\mathbf{F}\cdot \mathbf{T}, ds$$

Fundamental Theorem of Line Integrals: If $\mathbf{F} = \nabla f$, then

$$\int_C \mathbf{F} \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a))$$

If $C$ is closed,

$$\int_C \mathbf{F} \cdot d\mathbf{r} = 0$$

$\mathbf{F}$ is conservative if $\int_C \mathbf{F} \cdot d\mathbf{r}$ is path-independent. Path-independent $\iff$ $\mathbf{F} = \nabla f$ for some scalar function $f$ (potential function). If $\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$ is conservative $\iff$

$$\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}, \quad \frac{\partial P}{\partial z} = \frac{\partial R}{\partial x}, \quad \frac{\partial Q}{\partial z} = \frac{\partial R}{\partial y}$$

or $\mathbf{curl},\mathbf{F}=0$

Green's Theorem: positively oriented (counterclockwise), closed curve, region $D$ bounded by $C$:

$$\oint_{C}P,dx+Q,dy=\iint_{D}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA$$

$\int_{C}\mathbf{F}\cdot d\mathbf{r}$ if $\mathbf{F}$ is conservative ($Q_x=P_y$) $\iff$

$$\iint_{D}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA=0$$

Curl of $\mathbf{F}$ (takes vector):

$$\mathbf{curl},\mathbf{F} = \nabla\times\mathbf{F}=\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\mathbf{i}+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\mathbf{j}+\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\mathbf{k}$$

(returns vector)

Divergence of $\mathbf{F}$ (takes vector):

$$\mathbf{div},\mathbf{F} = \nabla\cdot\mathbf{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$$

(returns scalar)

Gradient of $\mathbf{f}$ (takes scalar):

$$\mathbf{grad},\mathbf{f} = \nabla\cdot\mathbf{f}=\langle f_x, f_y, f_z\rangle$$

(returns vector)

$$\mathbf{curl},(\mathbf{\nabla} f)=0 \qquad \mathbf{div},(\mathbf{curl},\mathbf{F})=0$$

Planes:

$$\mathbf{r}(u, v) = \mathbf{P}_0 + u\mathbf{a} + v\mathbf{b}$$

($\mathbf{a}$ and $\mathbf{b}$ are non-parallel vectors in the plane)

Explicit:

$$\mathbf{r}(u, v) = \langle u, v, f(u, v) \rangle$$

(if $z = f(x, y)$, $u=x$, $v=y$)

Spherical:

$$\mathbf{r}(\rho, \theta, \phi) = \langle \rho \sin \phi \cos \theta, \rho \sin \phi \sin \theta, \rho \cos \phi \rangle$$

(use if $x^2 + y^2 + z^2 = \rho^2$)

Cone:

$$\mathbf{r}(u, v) = \langle u \cos v, u \sin v, u \rangle$$

(use if $z = \pm \sqrt{x^2 + y^2}$)

Cylinder:

$$\mathbf{r}(\theta, z) = \langle \rho \cos \theta, \rho \sin \theta, z \rangle$$

(use if $x^2 + y^2 = \rho^2$ and $\rho$ is a constant radius)

Surface Integral of a Scalar Function (Surface Area Element $dS$):

$$A(S) = \iint_{D} |\mathbf{r}_u \times \mathbf{r}_v| , dA \quad \text{if parameterized by } \mathbf{r}(u, v)$$

OR

$$A(S) = \iint_{D} \sqrt{\left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 + 1} , dA \quad \text{if } z=f(x, y) \text{ (Explicit function)}$$

$$\iint_{S} f(x, y, z) , dS = \iint_{D} f(\mathbf{r}(u, v)) |\mathbf{r}_u \times \mathbf{r}_v| , dA$$

OR

$$\iint_{D} f(x, y, g(x, y)) \sqrt{\left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 + 1} , dA$$

OR

$$\iint_{D} f(x, h(x,z), z) \sqrt{\left(\frac{\partial y}{\partial x}\right)^2 + \left(\frac{\partial y}{\partial z}\right)^2 + 1} , dA$$

OR

$$\iint_{D} f(j(y,z), y, z) \sqrt{\left(\frac{\partial x}{\partial y}\right)^2 + \left(\frac{\partial x}{\partial z}\right)^2 + 1} , dA$$

Surface Integral of a Vector Field (Flux Integral $d\mathbf{S}$):

$$\iint_{S} \mathbf{F} \cdot d\mathbf{S} = \iint_{D} \mathbf{F}(\mathbf{r}(u, v)) \cdot (\mathbf{r}_u \times \mathbf{r}_v) , dA$$

OR

$$\iint_{S} \mathbf{F} \cdot d\mathbf{S} = \iint_{D} \left( -P \frac{\partial g}{\partial x} - Q \frac{\partial g}{\partial y} + R \right) , dA$$

($z=g(x,y)$, upward orientation (normal vector points up), and $\mathbf{F} = \langle P, Q, R \rangle$)

Stokes' Theorem:

$$\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} \mathbf{curl}, \mathbf{F} \cdot d\mathbf{S}$$

orientation of $C$ (boundary curve) and $\mathbf{n}$ (normal vector of $S$) must follow the Right-Hand Rule.

Divergence Theorem (for closed surface, oriented outward normal vector):

$$\iint_{S} \mathbf{F} \cdot d\mathbf{S} = \iiint_{E} \mathbf{div}, \mathbf{F} , dV = \iiint_{E} \nabla \cdot \mathbf{F} , dV$$