Calculus 3 – Comprehensive

Vectors in 2D & 3D

Vector Notation

2D: $\mathbf{v} = \langle v_1, v_2 \rangle$

3D: $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$

Magnitude

$|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$

Unit Vector

$\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}$

Vector Addition

$\mathbf{u} + \mathbf{v} = \langle u_1+v_1, u_2+v_2, u_3+v_3 \rangle$

Dot Product

Algebraic Form

$\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3$

Geometric Form

$\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta$

Angle Between Vectors

$\cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}$

Orthogonal Vectors

$\mathbf{u} \perp \mathbf{v}$ iff $\mathbf{u} \cdot \mathbf{v} = 0$

Projection

$\text{comp}_{\mathbf{v}}\mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|}$

Cross Product

Definition

$$\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$$

Magnitude

$|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin\theta$

Properties

$\mathbf{u} \times \mathbf{v}$ is perpendicular to both

Right-hand rule determines direction

Applications

Area of parallelogram: $A = |\mathbf{u} \times \mathbf{v}|$

Volume of parallelepiped: $V = |(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}|$

Lines & Planes in 3D

Parametric Line

$\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{d}$

$\mathbf{r}_0$ is point, $\mathbf{d}$ is direction

Symmetric Line

$\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$

Plane Equation

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

Normal vector: $\mathbf{n} = \langle a, b, c \rangle$

Distance from Point to Plane

$$d = \frac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2+b^2+c^2}}$$

Vector Functions & Curves

Definition

$\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle$

Derivative

$\mathbf{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$

Tangent vector to curve

Unit Tangent Vector

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

Integral

$\int \mathbf{r}(t) dt = \langle \int f(t)dt, \int g(t)dt, \int h(t)dt \rangle + \mathbf{C}$

Arc Length

$$s = \int_a^b |\mathbf{r}'(t)| dt$$

Curvature & TNB Frame

Curvature

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

Alt: $\kappa = \frac{|\mathbf{r}' \times \mathbf{r}''|}{|\mathbf{r}'|^3}$

Principal Normal

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

Binormal Vector

$\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)$

Radius of Curvature

$\rho = \frac{1}{\kappa}$

Torsion

$$\tau = -\frac{d\mathbf{B}}{ds} \cdot \mathbf{N}$$

Functions of Several Variables

Notation

$z = f(x,y)$ or $w = f(x,y,z)$

Limits

$\lim_{(x,y) \to (a,b)} f(x,y) = L$

Must approach from all directions

Continuity

$f$ continuous at $(a,b)$ if:

1) $f(a,b)$ exists

2) $\lim_{(x,y) \to (a,b)} f(x,y)$ exists

3) Limit equals $f(a,b)$

Level Curves & Surfaces

Level curve: $f(x,y) = c$

Level surface: $f(x,y,z) = c$

Partial Derivatives

First Order

$f_x = \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h}$

$f_y = \frac{\partial f}{\partial y} = \lim_{h \to 0} \frac{f(x,y+h) - f(x,y)}{h}$

Higher Order

$f_{xx} = \frac{\partial^2 f}{\partial x^2}$

$f_{xy} = \frac{\partial^2 f}{\partial y \partial x}$

Schwarz's Theorem

If $f_{xy}$ and $f_{yx}$ continuous:

$f_{xy} = f_{yx}$

Differentiability

If $f_x$ and $f_y$ exist and continuous: $f$ is differentiable

Chain Rule (Multivariable)

Case 1: $z = f(x,y)$, $x = x(t)$, $y = y(t)$

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

Case 2: $z = f(x,y)$, $x = x(s,t)$, $y = y(s,t)$

$$\frac{\partial z}{\partial s} = f_x \frac{\partial x}{\partial s} + f_y \frac{\partial y}{\partial s}$$

$$\frac{\partial z}{\partial t} = f_x \frac{\partial x}{\partial t} + f_y \frac{\partial y}{\partial t}$$

Implicit Differentiation

If $F(x,y) = 0$: $\frac{dy}{dx} = -\frac{F_x}{F_y}$

Directional Derivatives & Gradient

Gradient Vector

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

Directional Derivative

$D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}$

$\mathbf{u}$ must be unit vector

Properties of Gradient

Points in direction of max increase

Perpendicular to level curves

$D_{\mathbf{u}}f_{\max} = |\nabla f|$

Linear Approximation

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

Tangent Planes & Normal Lines

Tangent Plane to $z = f(x,y)$

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

Tangent Plane to Surface $F(x,y,z) = 0$

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

Normal Line

$$\frac{x-x_0}{F_x} = \frac{y-y_0}{F_y} = \frac{z-z_0}{F_z}$$

Normal = gradient direction

Extrema & Critical Points

Critical Points

Solve $\nabla f = \mathbf{0}$

$f_x = 0$ and $f_y = 0$

Second Derivative Test

Let $D = f_{xx}f_{yy} - (f_{xy})^2$

$D > 0$, $f_{xx} > 0$: local min

$D > 0$, $f_{xx} < 0$: local max

$D < 0$: saddle point

$D = 0$: test inconclusive

Absolute Extrema

Check critical points & boundaries

Lagrange Multipliers

Setup

Optimize $f(x,y,z)$ subject to $g(x,y,z) = c$

Condition

$\nabla f = \lambda \nabla g$

System to Solve

1) $f_x = \lambda g_x$

2) $f_y = \lambda g_y$

3) $f_z = \lambda g_z$

4) $g(x,y,z) = c$

Multiple Constraints

$\nabla f = \lambda_1 \nabla g_1 + \lambda_2 \nabla g_2$

Double Integrals

Rectangular Coordinates

$$\iint_R f(x,y) dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) dy dx$$

Polar Coordinates

$x = r\cos\theta$, $y = r\sin\theta$

$dA = r dr d\theta$

$$\iint_R f(r,\theta) r dr d\theta$$

Applications

Volume: $V = \iint_R f(x,y) dA$

Area: $A = \iint_R dA$

Mass: $M = \iint_R \rho(x,y) dA$

Use polar for circular regions

Triple Integrals

Rectangular Coordinates

$dV = dz dy dx$

Cylindrical Coordinates

$x = r\cos\theta$, $y = r\sin\theta$, $z = z$

$dV = r dz dr d\theta$

Spherical Coordinates

$x = \rho\sin\phi\cos\theta$

$y = \rho\sin\phi\sin\theta$

$z = \rho\cos\phi$

$dV = \rho^2 \sin\phi d\rho d\phi d\theta$

Jacobian & Change of Variables

2D Jacobian

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

Change of Variables

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

3D Jacobian

$$J = \begin{vmatrix} x_u & x_v & x_w \\ y_u & y_v & y_w \\ z_u & z_v & z_w \end{vmatrix}$$

Common Transformations

Polar: $J = r$

Cylindrical: $J = r$

Spherical: $J = \rho^2\sin\phi$

Line Integrals - Scalar Fields

Definition

$$\int_C f(x,y,z) ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)| dt$$

Arc Length

$$L = \int_C ds = \int_a^b |\mathbf{r}'(t)| dt$$

Mass of Wire

$$M = \int_C \rho(x,y,z) ds$$

Center of Mass

$$\bar{x} = \frac{1}{M} \int_C x \rho ds$$

Line Integrals - Vector Fields

Definition

$$\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) dt$$

Work

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

Work done by force field

Alternative Notation

$$\int_C P dx + Q dy + R dz$$

Independence of Path

Path independent if $\mathbf{F} = \nabla f$ (conservative)

Conservative Fields & Potential

Definition

$\mathbf{F}$ is conservative if $\mathbf{F} = \nabla f$

$f$ is the potential function

Test for Conservative (2D)

$\mathbf{F} = \langle P, Q \rangle$ is conservative if:

$$\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$$

Test for Conservative (3D)

$\mathbf{F} = \langle P, Q, R \rangle$ conservative iff:

$$\nabla \times \mathbf{F} = \mathbf{0}$$

FTC for Line Integrals

$$\int_C \nabla f \cdot d\mathbf{r} = f(B) - f(A)$$

Green's Theorem

Statement

$$\oint_C P dx + Q dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$$

Vector Form

$$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D (\nabla \times \mathbf{F}) \cdot \mathbf{k} dA$$

Flux Form

$$\oint_C \mathbf{F} \cdot \mathbf{n} ds = \iint_D \nabla \cdot \mathbf{F} dA$$

Area Formula

$$A = \frac{1}{2} \oint_C (x dy - y dx)$$

$C$ must be positively oriented closed curve

Surface Integrals - Scalar Fields

Definition

$$\iint_S f(x,y,z) dS = \iint_D f(\mathbf{r}(u,v)) |\mathbf{r}_u \times \mathbf{r}_v| du dv$$

Surface Area

$$S = \iint_S dS = \iint_D |\mathbf{r}_u \times \mathbf{r}_v| du dv$$

Graph of $z = f(x,y)$

$$dS = \sqrt{1 + f_x^2 + f_y^2} dA$$

Mass of Thin Surface

$$M = \iint_S \rho(x,y,z) dS$$

Surface Integrals - Vector Fields

Definition

$$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_S \mathbf{F} \cdot \mathbf{n} dS$$

Flux

$\iint_S \mathbf{F} \cdot d\mathbf{S}$ = flux of $\mathbf{F}$ through $S$

Upward Normal

$\mathbf{n} = \frac{(-f_x, -f_y, 1)}{\sqrt{1 + f_x^2 + f_y^2}}$

Computation

$$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F} \cdot (\mathbf{r}_u \times \mathbf{r}_v) du dv$$

Stokes' Theorem

Statement

$$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$$

Curl of Vector Field

$$\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}$$

Key Features

$C$ = boundary of $S$ (right-hand rule)

Circulation = flux of curl

Conservative fields: curl = 0

Divergence Theorem

Statement

$$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E \nabla \cdot \mathbf{F} dV$$

Divergence

$$\nabla \cdot \mathbf{F} = P_x + Q_y + R_z$$

Physical Meaning

Divergence = source/sink density

Flux outward = total divergence

Key Features

$S$ must be closed surface

$E$ is enclosed solid region

$\mathbf{n}$ points outward

Vector Operators & Identities

Gradient (Scalar to Vector)

$$\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle$$

Divergence (Vector to Scalar)

$$\nabla \cdot \mathbf{F} = P_x + Q_y + R_z$$

Curl (Vector to Vector)

$$\nabla \times \mathbf{F} = (R_y - Q_z)\mathbf{i} + (P_z - R_x)\mathbf{j} + (Q_x - P_y)\mathbf{k}$$

Laplacian

$$\nabla^2 f = f_{xx} + f_{yy} + f_{zz}$$

Important Identities

Identities

$\nabla \times (\nabla f) = \mathbf{0}$ (curl of gradient is zero)

$\nabla \cdot (\nabla \times \mathbf{F}) = 0$ (div of curl is zero)

$\nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla \cdot \mathbf{F}) - \nabla^2 \mathbf{F}$

Product Rules

$\nabla(fg) = f\nabla g + g\nabla f$

$\nabla \cdot (f\mathbf{F}) = f(\nabla \cdot \mathbf{F}) + \mathbf{F} \cdot \nabla f$

$\nabla \times (f\mathbf{F}) = f(\nabla \times \mathbf{F}) + (\nabla f) \times \mathbf{F}$

Useful Coordinate Conversions

Cylindrical to Rectangular

$x = r\cos\theta$

$y = r\sin\theta$

$z = z$

Spherical to Rectangular

$x = \rho\sin\phi\cos\theta$

$y = \rho\sin\phi\sin\theta$

$z = \rho\cos\phi$

Relationships

$r^2 = x^2 + y^2$

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

$\tan\theta = \frac{y}{x}$

Quick Reference: When to Use

Polar Coordinates

Circular/annular regions

Radially symmetric functions

Cylindrical Coordinates

Cylindrical or rotationally symmetric regions

Spherical Coordinates

Spherical regions

Functions depending on distance from origin

Green's Theorem

2D closed curve line integrals

Area calculations

Stokes' Theorem

Converting surface to line integrals

Divergence Theorem

Closed surface flux calculations

CALCULUS 3 Complete Reference