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
| Function f(x) | Derivative f'(x) |
|---|---|
| x^n | nx^{n-1} |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| e^x | e^x |
| ln(x) | 1/x |
| a^x | a^x ln(a) |
| tan(x) | sec²(x) |
| sec(x) | sec(x)tan(x) |
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
| Integral | Result |
|---|---|
| ∫x^n dx | x^{n+1}/(n+1) + C (n ≠ -1) |
| ∫1/x dx | ln |
| ∫e^x dx | e^x + C |
| ∫sin(x) dx | -cos(x) + C |
| ∫cos(x) dx | sin(x) + C |
| ∫sec²(x) dx | tan(x) + C |
| ∫dx/√(a²-x²) | arcsin(x/a) + C |
| ∫dx/(a²+x²) | (1/a) arctan(x/a) + C |
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
| Expression | Substitution | Result |
|---|---|---|
| √(a² - x²) | x = a sinθ | a cosθ |
| √(a² + x²) | x = a tanθ | a secθ |
| √(x² - a²) | x = a secθ | a tanθ |
Partial Fractions
For rational functions P(x)/Q(x):
- Divide if degree(P) ≥ degree(Q)
- Factor Q(x) into linear/quadratic factors
- Set up partial fraction decomposition
- 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
| Test | Application | Statement |
|---|---|---|
| nth-term | All series | lim a_n ≠ 0 → diverges |
| Geometric | a_n = ar^{n-1} | |r| < 1 converges |
| p-series | a_n = 1/n^p | p > 1 converges |
| Integral | a_n = f(n), f decreasing | ∫f converges ↔ series converges |
| Comparison | a_n ≤ b_n | b converges → a converges |
| Limit Comparison | lim a_n/b_n = L | Both converge or diverge |
| Ratio | lim | a_{n+1}/a_n |
| Root | lim sup | a_n |
| Alternating | a_n ≥ 0, decreasing | Alternating series converges |
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
| Operator | Definition | Output |
|---|---|---|
| ∇f | (∂f/∂x, ∂f/∂y, ∂f/∂z) | Vector |
| ∇·F | ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z | Scalar |
| ∇×F | (∂F_z/∂y - ∂F_y/∂z, ...) | Vector |
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
| Roots | Solution Form |
|---|---|
| Real, r₁, r₂ | y = C₁e^{r₁x} + C₂e^{r₂x} |
| Real repeated r | y = (C₁ + C₂x)e^{rx} |
| Complex α ± iβ | y = e^{αx}(C₁cosβx + C₂sinβx) |
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
| Equation | Solution | Applications |
|---|---|---|
| Bessel | J_n(x), Y_n(x) | Cylindrical symmetry |
| Legendre | P_n(x), Q_n(x) | Spherical symmetry |
| Hermite | H_n(x) | Quantum harmonic oscillator |
| Laguerre | L_n(x) | Hydrogen atom |
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