Calculus 3 Formulas

Vectors

Dot Product: \[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \] \[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \]

Projection of (\mathbf{a}) onto (\mathbf{b}): \[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b} \]

Cross Product: \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \ \end{vmatrix} \] \[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin \theta \]

Area of Parallelogram (in vectors): \[ \text{Area} = |\mathbf{a} \times \mathbf{b}| \]

Volume of Parallelepiped: \[ \text{Volume} = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| \]

Equation of a Line: \[ \mathbf{r} = \mathbf{r_0} + t \mathbf{v} \]

Parametric Equation of a Line: \[ \begin{cases} x = x_0 + t a \\ y = y_0 + t b \\ z = z_0 + t c \ \end{cases} \]

Symmetric Equation of a Line: \[ \frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} \]

Equation of a Plane (standard form): \[ ax + by + cz = d \]

Equation of a Plane (general form): \[ ax + by + cz + d = 0 \]

Distance from Point to Line: \[ D = \frac{|\mathbf{PQ} \times \mathbf{v}|}{|\mathbf{v}|} \]

Distance from Point to Plane: \[ D = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}} \]

Distance Between Parallel Planes: \[ D = \frac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}} \]

Position Vector for a Projectile: \[ \mathbf{r}(t) = \left( v_{0x} t \right) \mathbf{i} + \left( v_{0y} t - \frac{1}{2} g t^2 \right) \mathbf{j} \]

Unit Tangent Vector: \[ \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|} \]

Principal Unit Normal Vector: \[ \mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|} \]

Tangential and Normal Components of Acceleration: \[ a_T = \mathbf{a} \cdot \mathbf{T} \] \[ a_N = \sqrt{|\mathbf{a}|^2 - a_T^2} \]

Arc Length of a Space Curve: \[ s = \int_a^b |\mathbf{r}'(t)| , dt \]

Curvature: \[ \kappa = \frac{|\mathbf{T}'(t)|}{|\mathbf{r}'(t)|} = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3} \]

Angle of Inclination: \[ \theta = \arccos\left(\frac{a}{\sqrt{a^2 + b^2 + c^2}}\right) \]

Tangent Plane to Level Surface: \[ z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \]

Normal Line: \[ \frac{x - x_0}{f_x(x_0, y_0)} = \frac{y - y_0}{f_y(x_0, y_0)} = \frac{z - z_0}{-1} \]

Vector Fields: \[ \mathbf{F}(x, y, z) = P(x, y, z) \mathbf{i} + Q(x, y, z) \mathbf{j} + R(x, y, z) \mathbf{k} \]

Inverse Square Fields: \[ \mathbf{F}(\mathbf{r}) = \frac{k \mathbf{r}}{|\mathbf{r}|^3} \]

Conservative Vector Field Tests: \[ \nabla \times \mathbf{F} = \mathbf{0} \]

Curl of a Vector Field: \[ \nabla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \]

Divergence of a Vector Field: \[ \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]

Line Integrals: \[ \int_C f(x, y, z) , ds \]

Line Integrals of a Vector Field: \[ \int_C \mathbf{F} \cdot d\mathbf{r} \]

Differential Form of Line Integral: \[ \int_C P , dx + Q , dy + R , dz \]

Fundamental Theorem of Line Integrals: \[ \int_C \nabla f \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a)) \]

Independence of Path: \[ \int_C \mathbf{F} \cdot d\mathbf{r} = 0 \ \text{if} \ \mathbf{F} \ \text{is conservative} \]

Partial Derivatives

Total Differential: \[ dz = f_x , dx + f_y , dy \]

Chain Rule for Functions of One Independent Variable: \[ \frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt} \]

Chain Rule for Functions of Two Independent Variables: \[ \frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t} \]

Implicit Partial Differentiation: \[ \frac{\partial z}{\partial x} = -\frac{F_x}{F_z} \] \[ \frac{\partial z}{\partial y} = -\frac{F_y}{F_z} \]

Directional Derivative: \[ D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u} \]

The Gradient of a Function: \[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \]

Second Partials Test: \[ D = f_{xx} f_{yy} - (f_{xy})^2 \]

Extreme Value Theorem: \[ f \ \text{has a local maximum/minimum at} \ (a, b) \ \text{if} \ D > 0 \ \text{and} \ f_{xx} > 0 \ (\text{minimum}) \ \text{or} \ f_{xx} < 0 \ (\text{maximum}) \]

Lagrange Multipliers: \[ \nabla f = \lambda \nabla g \]

Iterated, Double, and Repeated Integrals

Rectangular Regions: \[ \int_a^b \int_c^d f(x, y) , dy , dx \]

Fubini's Theorem: \[ \int_a^b \int_c^d f(x, y) , dy , dx = \int_c^d \int_a^b f(x, y) , dx , dy \]

General Regions: \[ \int_{\text{Region}} f(x, y) , dA \]

Average Value of a Function Over a Region: \[ \frac{1}{\text{Area}} \int_{\text{Region}} f(x, y) , dA \]

Double Integrals in Polar Coordinates: \[ \int_a^b \int_{r_1(\theta)}^{r_2(\theta)} f(r, \theta) , r , dr , d\theta \]

Triple Integrals in Cylindrical Coordinates: \[ \int_a^b \int_{r_1(\theta)}^{r_2(\theta)} \int_{z_1(r, \theta)}^{z_2(r, \theta)} f(r, \theta, z) , r , dz , dr , d\theta \]

Triple Integrals in Spherical Coordinates: \[ \int_a^b \int_{0}^{\pi} \int_{0}^{2\pi} f(\rho, \phi, \theta) , \rho^2 \sin(\phi) , d\theta , d\phi , d\rho \]

Surface Area: \[ \iint_S |\mathbf{r}_u \times \mathbf{r}_v| , dA \]

Change of Variables for Double Integrals: \[ \iint_R f(x, y) , dx , dy = \iint_{R'} f(g(u, v), h(u, v)) \left| \frac{\partial(x, y)}{\partial(u, v)} \right| , du , dv \]

Triple Integrals: \[ \iiint_V f(x, y, z) , dV \]

Green's Theorem: \[ \oint_C (P , dx + Q , dy) = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \]

Alternate Forms of Green's Theorem: \[ \oint_C (L , dx + M , dy + N , dz) = \iint_R (\nabla \times \mathbf{F}) \cdot \mathbf{n} , dA \]

Parametric Surfaces: \[ \mathbf{r}(u, v) = \langle x(u, v), y(u, v), z(u, v) \rangle \]

Area of Parametric Surface: \[ \iint_D |\mathbf{r}_u \times \mathbf{r}_v| , du , dv \]

Surface Integrals: \[ \iint_S f(x, y, z) , dS \]

Flux Integrals: \[ \iint_S \mathbf{F} \cdot \mathbf{n} , dS \]

Divergence Theorem: \[ \iiint_V (\nabla \cdot \mathbf{F}) , dV = \iint_S \mathbf{F} \cdot \mathbf{n} , dS \]

Stokes' Theorem: \[ \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} , dS = \oint_C \mathbf{F} \cdot d\mathbf{r} \]

Surfaces

Ellipsoid: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \]

Hyperboloid of One Sheet: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \]

Hyperboloid of Two Sheets: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \]

Elliptic Cone: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0 \]

Elliptic Paraboloid: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = z \]

Hyperbolic Paraboloid (saddle): \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = z \]

Coordinate System Conversions

Cylindrical to Rectangular: \[ x = r \cos \theta \] \[ y = r \sin \theta \] \[ z = z \]

Spherical to Rectangular: \[ x = \rho \sin \phi \cos \theta \] \[ y = \rho \sin \phi \sin \theta \] \[ z = \rho \cos \phi \]

Spherical to Cylindrical: \[ r = \rho \sin \phi \] \[ \theta = \theta \] \[ z = \rho \cos \phi \]

Cylindrical to Spherical: \[ \rho = \sqrt{r^2 + z^2} \] \[ \phi = \arctan \left( \frac{r}{z} \right) \] \[ \theta = \theta \]