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Multivariable Calculus — Partial Derivatives and Gradients

Extending calculus to functions of multiple variables

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grad_school_grind 25 terms Mar 6, 2026
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Terms 25

1
Partial Derivative ∂f/∂x
Derivative of f with respect to x, treating all other variables as constants
2
Gradient ∇f
Vector of all partial derivatives: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z); points in direction of steepest ascent
3
Directional Derivative
Rate of change of f in direction of unit vector u: D_u f = ∇f · u
4
Critical Point
Point where all partial derivatives equal zero; candidate for local extrema
5
Second Derivative Test (multivariable)
Uses Hessian determinant D = f_xx·f_yy − (f_xy)²; D>0 and f_xx>0 → local min
6
Hessian Matrix
Square matrix of second-order partial derivatives; used to classify critical points
7
Saddle Point
Critical point where D < 0; neither max nor min; surface curves up in one direction and down in another
8
Chain Rule (multivariable)
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt) for z = f(x(t), y(t))
9
Implicit Differentiation (multivariable)
dy/dx = −(∂F/∂x)/(∂F/∂y) when F(x,y) = 0 defines y implicitly
10
Double Integral
∬f(x,y) dA; integrates over a 2D region; gives volume under surface
11
Fubini's Theorem
∬f(x,y) dA = ∫∫f(x,y)dx dy = ∫∫f(x,y)dy dx; order of integration can be switched under mild conditions
12
Triple Integral
∭f(x,y,z) dV; integrates over a 3D region
13
Jacobian
Determinant of matrix of partial derivatives; scales area/volume in change of variables
14
Polar Coordinates
x = r·cos(θ), y = r·sin(θ); dA = r·dr·dθ; useful for circular regions
15
Cylindrical Coordinates
Extends polar to 3D: (r, θ, z); dV = r·dr·dθ·dz
16
Spherical Coordinates
(ρ, θ, φ); dV = ρ²·sin(φ)·dρ·dθ·dφ; useful for spherical regions
17
Line Integral
∫_C f ds; integrates a function along a curve C in space
18
Vector Field
Function assigning a vector to each point in space; F(x,y,z) = (P,Q,R)
19
Conservative Vector Field
Field with potential function f such that ∇f = F; curl F = 0
20
Green's Theorem
∮_C (P dx + Q dy) = ∬(∂Q/∂x − ∂P/∂y) dA; relates line integral to double integral
21
Stokes' Theorem
∮_C F·dr = ∬_S (curl F)·dS; generalizes Green's theorem to 3D surfaces
22
Divergence Theorem (Gauss)
∯_S F·dS = ∭_V div F dV; flux through closed surface equals divergence over volume
23
Divergence
div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z; measures net outflow of vector field
24
Curl
curl F = ∇×F; measures rotation or circulation of vector field at a point
25
Lagrange Multipliers
Method for optimization with constraints: ∇f = λ∇g where g = 0 is the constraint