DerivativeCalculus.com

Multivariable Calculus Cheat Sheet

Your comprehensive guide to Calculus III: Partial derivatives, multiple integrals, vector calculus, and 3D concepts

2D
Single-Variable Calculus

Functions: f(x)

Derivative: f'(x)

Integral: ∫f(x)dx

3D
Multivariable Calculus

Functions: f(x,y,z)

Partial: ∂f/∂x

Integral: ∭f(x,y,z)dV

nD
Vector Calculus

Vector Fields: F(x,y,z)

Gradient: ∇f

Integral: ∮F·dr

∂ Partial Derivatives
∂ Partial Derivative (x)
∂f/∂x = limh→0 [f(x+h,y) - f(x,y)]/h
Notation:
fx, ∂f/∂x, Dxf
Example:
For f(x,y) = x²y + sin(x) ∂f/∂x = 2xy + cos(x)
∂² Second Order Partial
∂²f/∂x² = ∂/∂x(∂f/∂x)
∂²f/∂x∂y = ∂/∂y(∂f/∂x)
Clairaut's Theorem:
If fxy and fyx are continuous, then: fxy = fyx
∇ Gradient Vector
∇f = ⟨∂f/∂x, ∂f/∂y, ∂f/∂z⟩
Properties:
• Points in direction of greatest increase
• Perpendicular to level curves/surfaces
⛓️ Multivariable Chain Rule
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)
For z = f(x,y):
If x = g(t), y = h(t), then: dz/dt = fxg'(t) + fyh'(t)
∫∫ Multiple Integrals
∫∫ Double Integral
∬R f(x,y) dA
Iterated Form:
∫ab ∫g₁(x)g₂(x) f(x,y) dy dx
∫∫∫ Triple Integral
∭E f(x,y,z) dV
Volume Element:
dV = dz dy dx (in rectangular)
🔄 Change of Variables
∬R f(x,y) dA = ∬S f(g(u,v), h(u,v)) |J| dA
Jacobian:
J = ∂(x,y)/∂(u,v) = | ∂x/∂u ∂x/∂v | | ∂y/∂u ∂y/∂v |
↻ Polar Coordinates
x = r cos θ, y = r sin θ
dA = r dr dθ
Example:
∬circle f(x,y) dA = ∫02π ∫0R f(r cos θ, r sin θ) r dr dθ
⇀ Vector Calculus
∇· Divergence
div F = ∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
Interpretation:
Measures source/sink strength at a point
∇× Curl
curl F = ∇×F = ⟨ ∂Fz/∂y - ∂Fy/∂z, ∂Fx/∂z - ∂Fz/∂x, ∂Fy/∂x - ∂Fx/∂y ⟩
Interpretation:
Measures rotation/curl at a point
∇² Laplacian
∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²
Also:
∇²f = ∇·(∇f)
∫C Line Integral
∫C F·dr = ∫ab F(r(t))·r'(t) dt
For path C: r(t), a ≤ t ≤ b
Work done by force field F along path C
🗺️ Coordinate Systems
3D Coordinate Systems
📍 (x,y,z) → 🌐 (r,θ,φ)
⚓ Cylindrical Coordinates 3D
x = r cos θ, y = r sin θ, z = z
dV = r dz dr dθ
Best for:
• Cylinders
• Axial symmetry
🌐 Spherical Coordinates 3D
x = ρ sin φ cos θ
y = ρ sin φ sin θ
z = ρ cos φ
dV = ρ² sin φ dρ dφ dθ
Best for:
• Spheres
• Radial symmetry
📐 Transformation Jacobians
Rect→Cyl: J = r
Rect→Sph: J = ρ² sin φ
Remember:
dA or dV = |J| du dv dw
🚀 Key Applications & Theorems
📏 Tangent Plane
z - z₀ = fx(x₀,y₀)(x-x₀) + fy(x₀,y₀)(y-y₀)
Linear Approximation:
L(x,y) = f(x₀,y₀) + fx(x₀,y₀)(x-x₀) + fy(x₀,y₀)(y-y₀)
↗️ Directional Derivative
Duf = ∇f·u
Unit vector u:
u = ⟨a,b⟩/√(a²+b²)
🟢 Green's Theorem
∮C F·dr = ∬D (∂Q/∂x - ∂P/∂y) dA
For F = ⟨P,Q⟩:
Relates line integral around simple closed curve to double integral over region
🧊 Divergence Theorem
∯S F·dS = ∭E (∇·F) dV
Also called:
Gauss's Theorem
Relates flux through closed surface to divergence in volume
💡 Memory Tips & Mnemonics
Gradient Direction
"Up the Hill":
∇f points uphill in direction of steepest ascent
Negative ∇f points downhill
Curl Right-Hand Rule
Right Hand Rule:
Curl fingers in rotation direction, thumb points in curl vector direction
Polar Coordinates
"The r is extra":
dA = r dr dθ (always multiply by r!)
Don't forget the extra r!
Spherical Coordinates
"ρ² sin φ":
dV = ρ² sin φ dρ dφ dθ
Remember: ρ² and sin φ!