Formula sheet

Every formula from the course, organized by topic. Click a math expression to copy.

3D Space, Distance, Surfaces & Regions

4 formulas

Distance in 3D

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}

Distance between two points in 3-space.

Midpoint in 3D

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}, \frac{z _{1} + z _{2}}{2})

Midpoint of a segment in 3-space.

Sphere

(x - a)^{2} + (y - b)^{2} + (z - c)^{2} = r^{2}

Sphere with center (a,b,c) and radius r.

Circular cylinder along z-axis

x^{2} + y^{2} = r^{2}

When a variable is missing in 3D, the surface extrudes along that axis.

Vectors in the Plane and in Space

6 formulas

Vector magnitude

∣ v ∣ = v_{1}^{2} + v_{2}^{2} + v_{3}^{2}

Length of a 3D vector.

Vector addition

u + v = ⟨ u_{1} + v_{1}, u_{2} + v_{2}, u_{3} + v_{3} ⟩

Add componentwise.

Scalar multiple

c v = ⟨ c v_{1}, c v_{2}, c v_{3} ⟩

Scale each component by c.

Vector from points

P Q = ⟨ x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1} ⟩

Terminal minus initial.

Unit vector

\hat{v} = v /∣ v ∣

Direction with length 1.

2D vector from angle

v = ⟨ ∣ v ∣ cos θ, ∣ v ∣ sin θ ⟩

Build a vector from magnitude and direction.

Dot Product

6 formulas

Dot product (components)

u \cdot v = u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3}

Compute via components.

Dot product (geometric)

u \cdot v = ∣ u ∣∣ v ∣ cos θ

Use to extract the angle between vectors.

Angle between vectors

cos θ = \frac{u \cdot v}{∣ u ∣∣ v ∣}

Solve for θ ∈ [0, π].

Vector projection

proj_{v} u = \frac{u \cdot v}{∣ v ∣ ^{2}} v

Component of u along v as a vector.

Scalar projection

comp_{v} u = \frac{u \cdot v}{∣ v ∣}

Signed length of the projection.

Work (constant force)

W = F \cdot d

Constant force times displacement, dotted.

Cross Product

4 formulas

Cross product (components)

u \times v = ⟨ u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1} ⟩

Compute via component formula or determinant.

Cross product (determinant)

u \times v = i u_{1} v_{1} j u_{2} v_{2} k u_{3} v_{3}

Symbolic determinant form.

Cross product magnitude

∣ u \times v ∣ = ∣ u ∣∣ v ∣ sin θ

Area of parallelogram with sides u, v.

Triangle area

A = \frac{1}{2} ∣ u \times v ∣

Half of the parallelogram area.

Lines & Planes

6 formulas

Line: vector form

r (t) = r_{0} + t v

Parametric line through r0 with direction v.

Line: symmetric form

\frac{x - x _{0}}{a} = \frac{y - y _{0}}{b} = \frac{z - z _{0}}{c}

Equate ratios after eliminating t.

Plane: point-normal form

a (x - x_{0}) + b (y - y_{0}) + c (z - z_{0}) = 0

(a,b,c) is the normal vector.

Plane: standard form

a x + b y + cz = d

d = a x0 + b y0 + c z0.

Distance: point to plane

D = \frac{∣ a x _{0} + b y _{0} + c z _{0} - d ∣}{a ^{2} + b ^{2} + c ^{2}}

Standard formula.

Distance: point to line

D = \frac{∣ ( P - P _{0} ) \times v ∣}{∣ v ∣}

Uses cross product with direction.

Vector-Valued Functions

4 formulas

Line segment

r (t) = (1 - t) A + t B, 0 \leq t \leq 1

Parametric form for a segment from A to B.

Circle of radius a at (h,k)

r (t) = ⟨ h + a cos t, k + a sin t ⟩

Standard circle parametrization.

Ellipse with semi-axes a, b

r (t) = ⟨ a cos t, b sin t ⟩

Standard ellipse parametrization.

Helix

r (t) = ⟨ a cos t, a sin t, c t ⟩

Spiraling curve about z-axis.

Calculus of Vector Functions & Arc Length

5 formulas

Derivative

r^{'} (t) = ⟨ x^{'} (t), y^{'} (t), z^{'} (t)⟩

Componentwise differentiation.

Unit tangent

T (t) = r^{'} (t) /∣ r^{'} (t) ∣

Direction of motion, normalized.

Arc length

L = \int_{a}^{b} ∣ r^{'} (t) ∣ d t

Integral of speed.

Arc length element

d s = ∣ r^{'} (t) ∣ d t

Differential of arc length.

Tangent line

L (s) = r (t_{0}) + s r^{'} (t_{0})

Line tangent to curve at parameter $t_0$.

Functions of Several Variables

5 formulas

Level curve

f (x, y) = c

Set of points where f equals c.

Level surface

f (x, y, z) = c

Set in 3D where 3-variable f equals c.

Circular paraboloid

z = x^{2} + y^{2}

Upward; level curves are circles.

Saddle

z = x^{2} - y^{2}

Hyperbolic level curves.

Cone (upper half)

z = x^{2} + y^{2}

Apex at origin, opens upward.

Limits and Continuity

2 formulas

Substitution rule

(x, y) \to (a, b) lim f (x, y) = f (a, b)

If f continuous at (a,b).

Two-path test

γ_{1} lim f \neq = γ_{2} lim f \Rightarrow lim f DNE

Disagreement along any two paths kills the limit.

Partial Derivatives & Chain Rule

5 formulas

Partial derivative (limit definition)

f_{x} = h \to 0 lim \frac{f ( x + h , y ) - f ( x , y )}{h}

Definition of partial derivative.

Chain rule (one parameter)

\frac{d z}{d t} = f_{x} \frac{d x}{d t} + f_{y} \frac{d y}{d t}

z = f(x,y), x and y depend on t.

Chain rule (two parameters)

\frac{\partial z}{\partial s} = f_{x} x_{s} + f_{y} y_{s}

z = f(x,y), x and y depend on s, t.

Implicit differentiation

\frac{\partial z}{\partial x} = - \frac{F _{x}}{F _{z}}

When F(x,y,z) = 0 defines z implicitly.

Equality of mixed partials

f_{x y} = f_{y x}

If both are continuous (Clairaut).

Gradient, Directional Derivative, Tangent Planes

7 formulas

Gradient (3D)

\nabla f = ⟨ f_{x}, f_{y}, f_{z} ⟩

Partial derivatives packaged as a vector.

Directional derivative

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

u must be a unit vector.

Max rate of change

max D_{u} f = ∣\nabla f ∣

Achieved when u is along the gradient.

Tangent plane (z = f)

z = f (x_{0}, y_{0}) + f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0})

Linearization of z = f(x,y).

Tangent plane (level)

\nabla F (P_{0}) \cdot (r - r_{0}) = 0

Tangent plane to F(x,y,z) = k.

Normal line

r (t) = r_{0} + t \nabla F (r_{0})

Line perpendicular to a level surface at a point.

Differential

df = f_{x} d x + f_{y} d y

First-order change estimate.

Optimization

3 formulas

Critical points

f_{x} = 0, f_{y} = 0

Solve simultaneously.

Discriminant

D = f_{xx} f_{y y} - (f_{x y})^{2}

Used in second derivative test.

Lagrange multiplier condition

\nabla f = λ \nabla g

For constraint g(x, y, z) = c.

Double & Triple Integrals

4 formulas

Area

A = \iint_{R} d A

Region area as a double integral.

Volume (triple)

V = ∭_{E} d V

Solid volume.

Mass with density

m = \iint_{R} ρ d A

Or triple for solids.

Average value

\overset{ˉ}{f} = \frac{1}{A} \iint_{R} f d A

Average over the region.

Polar, Cylindrical & Spherical Coordinates

4 formulas

Polar conversion

x = r cos θ, y = r sin θ, d A = r d r d θ

2D conversions and area element.

Cylindrical conversion

x = r cos θ, y = r sin θ, z = z, d V = r d z d r d θ

3D, axial symmetry.

Spherical conversion

x = ρ sin ϕ cos θ, y = ρ sin ϕ sin θ, z = ρ cos ϕ

3D, radial.

Spherical volume element

d V = ρ^{2} sin ϕ d ρ d θ d ϕ

Critical Jacobian.

Vector Fields, Divergence & Curl

5 formulas

Divergence

\nabla \cdot F = P_{x} + Q_{y} + R_{z}

Scalar measure of outward flux density.

Curl

\nabla \times F = ⟨ R_{y} - Q_{z}, P_{z} - R_{x}, Q_{x} - P_{y} ⟩

Vector measure of rotation.

2D scalar curl

(curl F)_{z} = Q_{x} - P_{y}

Used in Green's theorem circulation form.

Conservative test

F conservative ⟺ \nabla \times F = 0 (simply connected)

On simply connected domains.

Potential

F = \nabla f

f is the potential function for F.

Line Integrals, Conservative Fields & Green's Theorem

7 formulas

Scalar line integral

\int_{C} f d s = \int_{a}^{b} f (r (t)) ∣ r^{'} (t) ∣ d t

Mass of a wire if f is density.

Vector line integral

\int_{C} F \cdot d r = \int_{a}^{b} F (r (t)) \cdot r^{'} (t) d t

Work / circulation along a curve.

Vector form (component)

\int_{C} P d x + Q d y = \int_{a}^{b} (P x^{'} + Q y^{'}) d t

Component-form alternative.

Fundamental Theorem of Line Integrals

\int_{C} \nabla f \cdot d r = f (B) - f (A)

Path-independent for conservative F.

Green's Theorem (circulation)

\oint_{C} P d x + Q d y = \iint_{D} (Q_{x} - P_{y}) d A

Closed-curve integral via curl.

Green's Theorem (flux)

\oint_{C} F \cdot n d s = \iint_{D} (P_{x} + Q_{y}) d A

Outward flux via divergence.

Area via Green's

A = \oint_{C} x d y

Pick P = 0, Q = x; gives Q_x - P_y = 1.

Surface Parameterization & Area

5 formulas

Surface area (parametric)

A = \iint_{D} ∣ r_{u} \times r_{v} ∣ d u d v

Magnitude of normal integrated over parameter domain.

Surface area (graph)

A = \iint_{R} 1 + f_{x}^{2} + f_{y}^{2} d A

For surfaces z = f(x,y).

Normal vector

n = r_{u} \times r_{v}

Direction depends on order of partials.

Sphere parametrization

r (θ, ϕ) = ⟨ a sin ϕ cos θ, a sin ϕ sin θ, a cos ϕ ⟩

Use spherical coordinates with fixed ρ = a.

Cylinder parametrization

r (θ, z) = ⟨ a cos θ, a sin θ, z ⟩

Polar in xy plus free z.