Calculus — Multivariable Functions, Partial Derivatives, and Multiple Integrals

The Basics of Multivariable Functions

Multivariable functions:

z = f(x, y): a two-variable function (in 3D space)
w = f(x, y, z): a three-variable function
Domain: a subset of the xy-plane

Limits and continuity (two variables):

lim f(x,y) = L: the same value results regardless of the path taken to (a,b)
Path dependence: if two paths give different limit values, the limit doesn’t exist
Continuity: lim f(x,y) = f(a,b)

Level curves:

f(x,y) = c (c constant): a curve in the x-y plane
Think of contour lines on a map, or isobars on a weather chart
Level surface: f(x,y,z) = c

Vector-valued functions:

r(t) = (x(t), y(t), z(t))
A parametric representation of a curve in 3D space
r’(t): the tangent vector (velocity vector)

Partial Derivatives and the Gradient

Partial derivative:

f_x = ∂f/∂x: the partial derivative with respect to x (treating y as constant)
f_y = ∂f/∂y: the partial derivative with respect to y (treating x as constant)

Example: f(x,y) = x²y + 3xy²

f_x = 2xy + 3y²
f_y = x² + 6xy

Higher-order partial derivatives:

f_xx = ∂²f/∂x²
f_xy = ∂²f/∂x∂y (a mixed partial derivative)
Clairaut’s theorem: f_xy = f_yx (assuming continuity)

The gradient:

∇f = (f_x, f_y) (two variables)
∇f = (f_x, f_y, f_z) (three variables)
Meaning: the direction and magnitude in which a function increases fastest
∇f ⊥ the level curve

Directional derivative:

D_u f = ∇f · u (in the direction of the unit vector u)
Maximum: in the direction of ∇f → magnitude |∇f|
Minimum: in the direction of −∇f → magnitude −|∇f|
Zero: when ∇f ⊥ u (along the level curve)

The chain rule:

For z = f(x,y), x = x(t), y = y(t): dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)
For x = x(s,t), y = y(s,t): ∂z/∂s = f_x · x_s + f_y · y_s

Classifying Extrema and Optimization

Critical points:
→ Points where ∇f = 0: f_x = 0 and f_y = 0
→ Can be a local maximum, local minimum, or saddle point

The second-derivative test:
→ D = f_xx · f_yy − (f_xy)²

D > 0, f_xx > 0: a local minimum
D > 0, f_xx < 0: a local maximum
D < 0: a saddle point
D = 0: inconclusive (further analysis needed)

Example: f(x,y) = x² + y² − 2x − 4y + 5
→ f_x = 2x − 2 = 0 → x = 1
→ f_y = 2y − 4 = 0 → y = 2
→ D = 2·2 − 0 = 4 > 0, f_xx = 2 > 0 → a local minimum

Lagrange multipliers:
→ Optimizing f(x,y) subject to the constraint g(x,y) = 0
→ ∇f = λ∇g (λ: the Lagrange multiplier)
→ The system of equations:
  f_x = λg_x
  f_y = λg_y
  g(x,y) = 0

Application: maximizing utility subject to a budget constraint
→ Maximize a utility function U(x,y)
→ Constraint: px·x + py·y = M (the budget)
→ Yields the condition: the marginal rate of substitution = the relative price ratio

Double Integrals

Double integral:

∬_R f(x,y) dA: an integral over region R
Geometric meaning: if z = f(x,y) ≥ 0, this gives a solid’s volume

Iterated integration (Fubini’s theorem):

∬_R f dA = ∫_a^b [∫_c^d f(x,y) dy] dx
The order of integration can be swapped (when f is continuous)

Type I region (simple in y):

a ≤ x ≤ b, g₁(x) ≤ y ≤ g₂(x)
∫a^b ∫{g1(x)}^(g2(x)) f dy dx

Type II region (simple in x):

c ≤ y ≤ d, h₁(y) ≤ x ≤ h₂(y)
∫c^d ∫{h1(y)}^(h2(y)) f dx dy

Polar coordinate conversion:

x = r cosθ, y = r sinθ
dA = r dr dθ
Useful for circular regions

The Jacobian:

Converts the area element under a coordinate change
x = x(u,v), y = y(u,v): J = | x_u x_v | | y_u y_v |
dA = |J| du dv

Applications of double integrals:

Area: ∬_R 1 dA
Volume: ∬_R f(x,y) dA (f ≥ 0)
Mass: ∬_R ρ(x,y) dA (density ρ)
Center of mass: x̄ = (1/m) ∬ x·ρ dA

Triple Integrals

Triple integral:

∭_E f(x,y,z) dV
Volume: ∭_E 1 dV

Cylindrical coordinates:

x = r cosθ, y = r sinθ, z = z
dV = r dr dθ dz
Convenient for cylindrical or conical regions

Spherical coordinates:

x = ρ sinφ cosθ
y = ρ sinφ sinθ
z = ρ cosφ
ρ: distance from the origin, φ: polar angle, θ: azimuthal angle
dV = ρ² sinφ dρ dφ dθ
Convenient for spherical regions

Applications:

Volume: ∭_E 1 dV
Mass: ∭_E ρ(x,y,z) dV
The volume of a sphere = (4/3)πR³ (derivable using spherical coordinates)

Frequently Asked Questions

Q. How does a partial derivative differ from a total derivative? A. A partial derivative ∂f/∂x is the rate of change when y is held fixed and only x varies. A total derivative df = f_x dx + f_y dy captures f’s total change when both x and y vary simultaneously. For a firm’s profit function f(price, sales volume), the partial derivative with respect to price is the profit change rate when sales volume is held fixed, while the total derivative approximates the change in profit when both variables move together.

Q. Is it sometimes easier to swap the order of integration? A. Yes — depending on the integrand or the shape of the region, one order can be far easier to compute. For example, ∫₀¹ ∫_x^1 sin(y²) dy dx can’t be integrated with respect to x directly for sin(y²), but swapping the order to ∫₀¹ ∫₀^y sin(y²) dx dy = ∫₀¹ y·sin(y²) dy converts it into something solvable by substitution.