Calculus

name: calculus

description: > Expert calculus assistant for mathematicians, scientists, and engineers. Use this skill whenever the user needs: help with differentiation and integration, solving differential equations, applying Taylor series, understanding multivariable calculus, working with vector calculus, or analyzing convergence of series. Includes both fundamental techniques and advanced applications. trigger: Any mathematical problem involving rates of change, accumulation, optimization, or mathematical modeling where calculus provides the analytical framework. license: MIT compatibility: opencode metadata: audience: mathematicians category: mathematics

Calculus — Differential and Integral Methods

Covers: Single Variable Calculus · Multivariable Calculus · Differential Equations · Series · Vector Calculus

Foundational Concepts

The Derivative

The derivative measures instantaneous rate of change:

f'(x) = lim_{h→0} [f(x+h) - f(x)]/h = df/dx

Geometrically: slope of tangent line at point.

Basic Differentiation Rules

Product Rule

(uv)' = u'v + uv'

Quotient Rule

(u/v)' = (u'v - uv')/v²

Chain Rule

(f∘g)'(x) = f'(g(x)) · g'(x)

In Leibniz notation:

dy/dx = (dy/du)(du/dx)

Implicit Differentiation

For F(x,y) = 0:

dy/dx = -∂F/∂x / ∂F/∂y

Example: x² + y² = r²

2x + 2y dy/dx = 0 → dy/dx = -x/y

Higher Derivatives

f''(x) = d²f/dx²
f^(n)(x) = d^n f/dx^n

Integration Fundamentals

The Integral as Antiderivative

∫f(x)dx = F(x) + C

Where F'(x) = f(x)

Fundamental Theorem of Calculus

If F is any antiderivative of f:

∫_a^b f(x)dx = F(b) - F(a)

This connects differentiation and integration as inverse operations.

Basic Integration Formulas

Integration by Parts

∫u dv = uv - ∫v du

LIATE rule: Choose u in order: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential

Example: ∫x e^x dx

• u = x, dv = e^x dx
• du = dx, v = e^x
• = x e^x - ∫e^x dx = x e^x - e^x + C

Trigonometric Integrals

Powers of sin and cos: Use identities:

• sin²x = (1 - cos(2x))/2
• cos²x = (1 + cos(2x))/2
• sin x cos x = sin(2x)/2

For odd powers: Save one factor, convert rest using sin²x + cos²x = 1

Trigonometric Substitution

Partial Fractions

For rational functions P(x)/Q(x):

1. Divide if degree(P) ≥ degree(Q)
2. Factor Q(x) into linear/quadratic factors
3. Set up partial fraction decomposition
4. Solve for coefficients

Rationalizing Substitutions

For integrals involving √{ax + b}: let u = √{ax + b}

Applications of Integration

Area Under Curve

A = ∫_a^b f(x)dx (f(x) ≥ 0)

Between two curves:

A = ∫_a^b |f(x) - g(x)| dx

Volume of Revolution

Washer method (around x-axis):

V = π∫_a^b [f(x)]² dx

Shell method (around y-axis):

V = 2π∫_a^b x f(x) dx

Arc Length

For y = f(x):

s = ∫_a^b √[1 + (f'(x))²] dx

Surface Area of Revolution

Around x-axis:

S = 2π∫_a^b f(x) √[1 + (f'(x))²] dx

Center of Mass

For region with density ρ(x):

x̄ = (∫_a^b xρ(x)f(x)dx) / (∫_a^b ρ(x)f(x)dx)

Work Problems

W = ∫F(x)dx

Examples:

• Spring: W = ½kx²
• Lifting: W = ∫mg dy
• Pumping: W = ∫ρg A(y) dy

Sequences and Series

Convergence Tests

Power Series

Series of form:

Σ a_n (x - c)^n

Radius of convergence (Cauchy-Hadamard):

R = 1/limsup |a_n|^{1/n}

Taylor Series

Taylor expansion about x = a:

f(x) = Σ f^{(n)}(a)/n! · (x - a)^n

Maclaurin series: Taylor series about x = 0

Common expansions:

e^x = Σ x^n/n! = 1 + x + x²/2! + ...
sin x = Σ (-1)^n x^{2n+1}/(2n+1)! 
cos x = Σ (-1)^n x^{2n}/(2n)!
ln(1+x) = Σ (-1)^{n+1} x^n/n  (|x| < 1)
(1+x)^α = Σ C(α,n) x^n

Remainder Estimation

For Taylor series, remainder after n terms:

|R_n(x)| ≤ M|x-a|^{n+1}/(n+1)!

Fourier Series

For periodic function f(x) with period 2L:

f(x) = a₀/2 + Σ [a_n cos(nπx/L) + b_n sin(nπx/L)]

Coefficients:

a_n = (1/L) ∫_{-L}^L f(x) cos(nπx/L) dx
b_n = (1/L) ∫_{-L}^L f(x) sin(nπx/L) dx

Multivariable Calculus

Partial Derivatives

For f(x,y):

∂f/∂x = lim_{h→0} [f(x+h,y) - f(x,y)]/h
∂f/∂y = lim_{h→0} [f(x,y+h) - f(x,y)]/h

Chain Rule (Multivariable)

For z = f(x,y), x = g(t), y = h(t):

dz/dt = ∂f/∂x · dx/dt + ∂f/∂y · dy/dt

For z = f(x,y), x = g(u,v), y = h(u,v):

∂z/∂u = ∂f/∂x ∂x/∂u + ∂f/∂y ∂y/∂u
∂z/∂v = ∂f/∂x ∂x/∂v + ∂f/∂y ∂y/∂v

Gradient

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)

Direction of steepest ascent; magnitude = maximum directional derivative.

Directional Derivative

D_uf = ∇f · û

Where û is unit vector in direction.

Multiple Integrals

Double integral over region R:

∬_R f(x,y) dA

Triple integral:

∭_V f(x,y,z) dV

Change of variables with Jacobian:

∭_V f dV = ∭_{V'} f(x(u,v,w),y(...),z(...)) |J| du dv dw

Common coordinate systems:

• Polar: dA = r dr dθ
• Cylindrical: dV = r dr dθ dz
• Spherical: dV = r² sinφ dr dφ dθ

Line Integrals

For vector field F and curve C:

∫_C F · dr = ∫ F · T ds

For scalar field:

∫_C f ds = ∫ f |r'(t)| dt

Surface Integrals

For surface S with normal n̂:

∬_S F · n̂ dS = ∬_D F(r(u,v)) · (r_u × r_v) du dv

Vector Calculus

Gradient, Divergence, and Curl

Identities

∇·(∇×F) = 0 (divergence of curl = 0)
∇×(∇f) = 0 (curl of gradient = 0)
∇·(∇f) = ∇²f (Laplacian)

Green's Theorem (2D)

∮_C P dx + Q dy = ∬_R (∂Q/∂x - ∂P/∂y) dA

Stokes' Theorem

∮_C F · dr = ∬_S (∇×F) · n̂ dS

Divergence Theorem

∬_S F · n̂ dS = ∭_V ∇·F dV

Fundamental Theorem for Line Integrals

For conservative field F = ∇f:

∫_C F · dr = f(r_end) - f(r_start)

Path-independent for simply connected regions.

Differential Equations

First-Order ODEs

Separable:

dy/dx = g(x)h(y)
∫ dy/h(y) = ∫ g(x) dx

Linear (integrating factor):

dy/dx + P(x)y = Q(x)
μ = exp(∫P dx)
d/dx(μy) = μQ

Exact:

M(x,y)dx + N(x,y)dy = 0
∂M/∂y = ∂N/∂x

Second-Order Linear ODEs

Homogeneous with constant coefficients:

ay'' + by' + cy = 0

Characteristic equation:

ar² + br + c = 0

Cauchy-Euler Equations

Form: ax²y'' + bxy' + cy = 0 Try solution y = x^r → characteristic equation.

Power Series Solutions

For y'' + P(x)y' + Q(x)y = 0:

y = Σ a_n x^n

Find recurrence relation for coefficients.

Special Functions

Laplace Transform

L{f(t)} = F(s) = ∫_0^∞ e^{-st} f(t) dt

Transforms:

• L{f'} = sF(s) - f(0)
• L{f''} = s²F(s) - sf(0) - f'(0)
• L{∫_0^t f(τ)dτ} = F(s)/s

Solving PDEs

Separation of variables: Assume u(x,t) = X(x)T(t) Fourier series: Expand initial conditions in eigenfunctions Transform methods: Laplace (time), Fourier (space)

Series Solutions and Special Cases

Convergence of Power Series

Absolute convergence: If Σ|a_n| converges Conditional convergence: Converges but not absolutely

Ratio test:

lim |a_{n+1}/a_n| = L
L < 1 → converges
L > 1 → diverges
L = 1 → inconclusive

Interval of convergence: Always centered at expansion point; test endpoints separately.

Operations on Power Series

• Differentiate term-by-term within radius of convergence
• Integrate term-by-term
• Add/subtract: combine coefficients
• Multiply: convolution of coefficients
• Compose: more complex, requires substitution

Asymptotic Series

Series that approximate function as parameter → ∞ or → 0:

f(x) ~ a₀ + a₁/x + a₂/x² + ...

Useful when ordinary series diverge.

Common Errors to Avoid

• Forgetting constant of integration C
• Applying chain rule incorrectly
• Using wrong sign in integration by parts
• Confusing converge/diverged in series tests
• Forgetting to check endpoints in interval of convergence
• Mixing up gradient (vector) with partial derivatives
• Forgetting Jacobian in change of variables
• Incorrectly applying product rule in vector derivatives
• Confusing curl and divergence physically
• Forgetting initial/boundary conditions in ODEs/PDEs

Key References

• Calculus by Spivak — Rigorous introduction
• Principles of Mathematical Analysis by Rudin — Advanced calculus
• Advanced Engineering Mathematics by Kreyszig — Applied perspective
• Mathematical Methods for Physicists by Arfken — Physics applications