Overview of Multivariable Calculus
Multivariable calculus extends single-variable calculus to functions of two or more
variables. We begin with vectors and geometry in 2D/3D, then study functions
f (x, y, … ), their limits and derivatives, and how to approximate and optimize them.
Next we cover multiple integrals (double, triple) and coordinate transforms (polar,
cylindrical, spherical). We then study vector fields – functions that assign a vector to
each point in space – and the calculus of vector fields: line and surface integrals, and the
fundamental theorems of Green, Gauss (divergence), and Stokes. Advanced topics
include the multivariable Taylor series (using the Hessian), Lagrange multipliers, and a
brief look at differential forms which generalize these ideas.
Roadmap: After covering the basics of vectors and coordinate systems, we discuss
partial derivatives and the gradient. We introduce the Jacobian and Hessian matrices for
changes of variables and second derivatives. Then we do double and triple integrals,
including applications like computing volumes and masses. Next are vector fields and
their operations (divergence, curl), followed by line and surface integrals. We present
Green’s Theorem in the plane, then the Divergence Theorem and Stokes’ Theorem in
space. Throughout we note physical applications (fluid flow, electromagnetism,
optimization). A table of contents of topics is given at the end, with examples, diagrams,
and recommended resources.
Figure: A saddle-shaped surface (the graph of f (x, y) = 12 cos(x/2) + sin(y/4))
​
showing a saddle point. Multivariable functions can have maxima, minima, or saddles; here
the surface curves up in one direction and down in another (saddle behavior)
en.wikipedia.org
.
Multivariable surfaces can be visualized by their contour lines (level sets) or by 3D
graphs like this. The gradient of f at each point is a vector normal to the surface’s level
curve there
en.wikipedia.org
.
1. Vectors and Coordinate Geometry
A vector in Rn is an ordered list of components (x1 , … , xn ) that can represent a point
​
​
or a directed displacement. In physics, vectors describe quantities like velocity, force, or
electric field. Key operations are addition and scalar multiplication (combining and
scaling vectors), and two products: the dot product a ⋅ b = a1 b1 + ⋯ + an bn , which
​
​
​
​
gives ∥a∥∥b∥ cos θ (so we can find angles or project one vector onto another), and the
cross product in R3 : a × b yields a new vector perpendicular to both, with magnitude
∥a∥∥b∥ sin θ. From these we get formulas for lengths (∥v∥ =
v12 + v22 + v32 ) and
​
​
​
angles. Planes and lines in 3D are defined by vectors and dot products (e.g. a plane
normal n satisfies n ⋅ (r − r0 ) = 0). Coordinates: a point (x, y, z) can be converted
​
to cylindrical (r, θ, z) or spherical (ρ, ϕ, θ) coordinates, where x = r cos θ, y =
r sin θ; x = ρ sin ϕ cos θ, y = ρ sin ϕ sin θ, z = ρ cos ϕ. These are used to simplify
integration over circular or spherical regions.
Example: The line through P0 = (1, 2, 3) in the direction of vector v = (4, −1, 2) is
​
given parametrically by r(t) = P0 + t v = (1 + 4t, 2 − t, 3 + 2t). The plane with
​
normal n = (2, −1, 2) through P0 has equation 2(x − 1) − (y − 2) + 2(z − 3) =
​
0.
Application: Vectors model forces and velocities in mechanics, making geometry in 3D
coordinate systems essential in physics and engineering. (For instance, the dot product
computes work: W = F ⋅ Δr.)
2. Vector-valued Functions (Parametric Curves)
A vector-valued function r(t) = (x(t), y(t), z(t)) describes a curve in space as t
varies. Its derivative r′ (t) gives the velocity vector (tangent) and ∥r′ (t)∥ is speed. The
arc length from t = a to t = b is s = ∫
b
​
∥r′ (t)∥ dt. Curvature quantifies bending: for
a
∣x′ y ′′ −y ′ x′′ ∣
a plane curve κ = ( x′2 +y ′2 )3/2 , or more generally using derivatives of r(t).
​
Example: Consider a helix r(t) = (cos t, sin t, t). Then r′ (t) = (− sin t, cos t, 1).
(− sin t)2 + (cos t)2 + 12 =
while moving upward. Curvature κ = 1/2.
The speed is
​
2. The curve winds around the z -axis
​
Visualization: These parametric curves trace trajectories of particles. In physics and
engineering, parametric curves model objects’ paths (e.g. projectile motion) or the shape
of springs and coils.
3. Functions of Several Variables
A multivariable function f (x, y) assigns a real number to each point (x, y) in its
domain. Its graph is a surface in R3 . More generally f (x, y, z) has a 4D graph (often
visualized via level surfaces). We visualize f (x, y) via level curves (contour lines
f (x, y) = c) or color plots. For example, f (x, y) = x2 + y 2 has level curves x2 +
y 2 = c (circles), and its graph is a paraboloid.
A level set f (x, y) = c is a curve (in 2D) or surface (in 3D) where f is constant. The
gradient of f , denoted ∇f , points in the direction of greatest increase of f and is
orthogonal to level sets
en.wikipedia.org
. (In fact, if F (x, y, z) = c defines a surface, then
∇F is a normal vector to that surface.)
Illustration: A saddle-shaped surface (graph of f (x, y) = 12 cos(x/2) + sin(y/4)). The
​
level curves of f would look like curved lines on the surface. The gradient ∇f at any point
is perpendicular to such a level curve
en.wikipedia.org
. Saddle points like this are neither
maxima nor minima of f (the surface curves up one way and down another).
Example: Let f (x, y) = x2 − y 2 . Its graph is a saddle. The level curves x2 − y 2 = c
are hyperbolas. The gradient is ∇f = (2x, −2y). At the origin ∇f = (0, 0), a critical
(stationary) point which is a saddle (not an extremum).
Topology & Continuity: Multivariable limits are more subtle: the limit of f (x, y) as
(x, y) → (a, b) must be the same along every path. For example, lim(x,y)→(0,0) x2xy
2 +y 2
is 0 along y = x but undefined (or differs) along y = 0, so the limit does not exist.
Continuity means f (x, y) approaches f (a, b) from all directions. These foundations
​
ensure well-defined graphs and level sets.
4. Partial Derivatives and the Gradient
​
∂f
The partial derivative ∂x measures the rate of change of f (x, y, … ) as x changes
​
(holding other variables fixed). For f (x, y), we define fx (x, y) =
​
(x,y)
limh→0 f (x+h,y)−f
and similarly fy . These exist under mild differentiability
h
​
​
conditions. The gradient is the vector of partials:
∇f (x, y) = (fx (x, y), fy (x, y)).
​
​
This is a vector field. Its magnitude and direction have meaning: ∥∇f ∥ is the maximum
rate of increase of f , and ∇f points where f increases fastest
en.wikipedia.org
. More
precisely, the directional derivative of f in a unit direction u is
f (x + hu) − f (x)
= ∇f (x) ⋅ u.
h→0
h
Du f = lim
​
​
​
So Du f = ∇f ⋅ u. In particular, the directional derivative is largest when u is the
​
direction of ∇f , and its magnitude equals ∥∇f ∥
en.wikipedia.org
en.wikipedia.org
.
Example: For f (x, y) = x2 + y 2 , fx = 2x, fy = 2y , so ∇f = (2x, 2y). At the point
​
​
(1, 2), ∇f = (2, 4) points toward increasing x and y . A unit vector in its direction is
u = (1/ 5, 2/ 5). The directional derivative in that direction is ∇f ⋅ u = (2, 4) ⋅
(1/ 5, 2/ 5) = ∥(2, 4)∥ (i.e.\ 2 5), the maximum rate of change there.
​
​
​
​
​
Application: The gradient is fundamental in optimization: stationary points (where
∇f = 0) are potential minima or maxima en.wikipedia.org . In physics, if f (x, y, z) is a
scalar potential (like temperature T or gravitational potential), then −∇f gives the force
or flow direction (heat flows downhill, objects fall along the negative gradient). For
example, water flows from high to low altitude, so ∇ of a height function shows
steepest ascent ( uphill)
en.wikipedia.org
.
5. The Multivariable Taylor Series and Hessian
Just as single-variable functions have Taylor polynomials, a smooth function f (x) has a
multivariable Taylor expansion around a point a. To second order,
f (x) ≈ f (a) + ∇f (a) ⋅ (x − a) + 12 (x − a)T Hf (a) (x − a),
​
​
where Hf (a) is the Hessian matrix (matrix of second partials) at a. For a function f :
​
2
f
. The Hessian appears
Rn → R, Hf is the n × n matrix with (Hf )ij = ∂x∂i ∂x
j
​
​
​
​
​
​
naturally: it generalizes the single-variable second derivative. In fact, the quadratic term
of the Taylor expansion can be written as 12 (x − a)T Hf (a)(x − a)
​
​
mathinsight.org
.
Example: Take f (x, y) = x2 + xy + y 2 . Then ∇f = (2x + y, x + 2y) and
Hf = (
​
2 1
).
1 2
​
Around (0, 0) the second-order Taylor is f (x, y) ≈ 0 + (2x + y, x + 2y) ⋅ (x, y) +
2
1
T =
(x,
(x,
y)H
y)
0
+
0
+
y)
(
(x,
f
2
1
​
​
​
1
) (x, y)T /2 = x2 + xy + y 2 , which
2
​
matches f exactly in this quadratic case.
Optimization: The Hessian is used in the second derivative test for critical points. If
∇f (a) = 0, then the eigenvalues of Hf (a) tell us the nature of a: if all eigenvalues are
positive (Hessian positive definite), a is a local minimum; if all negative, a local
maximum; if mixed signs, a saddle. (For the example Hf above, eigenvalues are 3 and 1,
so (0, 0) is a minimum.)
​
​
6. The Chain Rule
For composite multivariable functions, the chain rule lets us differentiate through the
composition. For example, if z = f (x, y), where x = g(t) and y = h(t), then
dz
= fx (g(t), h(t)) g ′ (t) + fy (g(t), h(t)) h′ (t).
dt
​
​
​
More generally for f (x, y, z) with each variable a function of t, df /dt = fx x′ +
​
fy y ′ + fz z ′ . In even higher dimensions, one uses the Jacobian matrix (matrix of first
​
​
partials) times the derivative vector. The chain rule is essential in dynamics and
geometry: for instance, when changing coordinates or when motion depends on several
variables.
7. Multiple Integration
Double Integrals
The double integral of a function f (x, y) over a region D ⊂ R2 , ∬D f (x, y) dA, gives
​
volumes (if f ≥ 0), mass (with density f ), or accumulated quantities. It can be
computed by iterated integrals (Fubini’s theorem):
∬ f (x, y ) dA = ∫
​
D
b
∫
g2 (x)
​
​
​
x=a y=g1 (x)
f (x, y) dy dx,
​
where D = {(x, y) : a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x)}. Integrals over more general
​
​
D or in the other order (dy dx vs dx dy ) can also be done.
Example: Compute the area of the unit circle x2 + y 2 ≤ 1. In Cartesian coordinates it is
∬x2 +y2 ≤1 1 dA. Using polar coordinates x = r cos θ, y = r sin θ (so dA = r dr dθ),
​
we get
2π
1
∫θ=0 ∫r=0 r dr dθ = 2π ⋅ 12 = π .
​
​
​
Polar, Cylindrical, Spherical Coordinates
Polar coordinates (r, θ ) are used in the plane: x = r cos θ, y = r sin θ . The Jacobian
is r : dA = r dr dθ . For triple integrals, cylindrical coords (r, θ, z) use x =
r cos θ, y = r sin θ, z = z (volume element r dr dθ dz ). Spherical coords (ρ, ϕ, θ)
use x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ with volume element
ρ2 sin ϕ dρ dϕ dθ. These often simplify integrals over circular/ spherical regions.
Example: Volume of a sphere radius R by triple integral in spherical:
2π
π
R
∫θ=0 ∫ϕ=0 ∫ρ=0 ρ2 sin ϕ dρ dϕ dθ = 43 πR3 .
​
​
​
​
8. Vector Fields and Vector Operators
A vector field F(x) assigns a vector to each point x in space (or plane). For example,
F(x, y, z) might describe the velocity of a fluid at each point, or the force of a
gravitational/magnetic field. Formally, F(x) : Rn → Rn . It can be visualized by arrows
drawn at points, showing direction and magnitude (like wind or current patterns). In
practice, physics and engineering use vector fields all the time: e.g. v(x, y, z) for fluid
flow, E or B for electric/magnetic fields.
Key operations on vector fields include:
∂
∂F
∂Fy
Divergence: ∇ ⋅ F = ∂xx + ∂y + ⋯. Divergence is a scalar field measuring
“outflowingness” (expansion) of F at each point math.libretexts.org . If F is a fluid velocity,
​
​
​
​
∇ ⋅ F > 0 means net flow out of a tiny volume (source), ∇ ⋅ F < 0 means net
inflow (sink), and zero means incompressible (volume-preserving) flow
math.libretexts.org
. Example: F = (x, y, z) has ∇ ⋅ F = 1 + 1 + 1 = 3, meaning it is diverging
outward (each point looks like a source).
Curl: ∇ × F (in 3D) is a vector measuring rotation or “circulation” at each point
math.libretexts.org
. More precisely, the curl’s magnitude equals twice the local angular
velocity of fluid rotation, and its direction is the axis of rotation. In Cartesian form,
for F = ⟨P , Q, R⟩:
∇×F=(
math.libretexts.org
∂R ∂Q ∂P
∂R ∂Q ∂P
) i, j, k
−
,
−
,
−
∂y
∂z ∂z
∂x ∂x
∂y
​
​
​
​
​
​
. If ∇ × F = 0 everywhere in a simply-connected domain, then F is
conservative (see below). Example: The field F(x, y, z) = (y, −x, 0) has constant
curl ∇ × F = (0, 0, −2), indicating a uniform rotation about the z -axis
math.libretexts.org
en.wikipedia.org
.
9. Line and Surface Integrals
Line Integrals of Scalar Fields
The line integral of a scalar function f (x, y, z) along a curve C (with arc-length
b
parameter s) is ∫C f ds. If r(t) parametrizes C , then ∫C f ds = ∫a f (r(t)) ∥r′ (t)∥ dt
​
​
​
. This computes, for example, mass on a wire of varying density f .
Line Integrals of Vector Fields (Work)
For a vector field F(r), the line integral ∫
​
F ⋅ dr represents the work done by force F
C
along the curve C . If r(t) parametrizes C from t = a to t = b, then
b
∫ F ⋅ dr = ∫ F(r(t)) ⋅ r′ (t) dt.
​
C
​
a
A fundamental result is that if F = ∇f is a conservative (gradient) field, then the line
integral depends only on the endpoints:
∫ ∇f ⋅ dr = f (end) − f (start),
​
C
regardless of path
. In other words, a continuous field is conservative if and
en.wikipedia.org
only if it is the gradient of some scalar potential, and its line integrals around closed
loops are zero
en.wikipedia.org
. This is the gradient (or potential) theorem for line integrals.
Example: Let F(x, y) = (2x, −3y). Then one potential is f (x, y) = x2 − 32 y 2 (since
​
∇f = (2x, −3y)). The work from (0, 0) to (1, 2) is f (1, 2) − f (0, 0) = (12 − 32 ⋅
4) − 0 = −5. Any other path yields the same work because F is conservative.
​
Surface Integrals and Flux
The surface integral of a scalar function or a vector field is defined over a surface S . In
particular, the flux of a vector field F through S measures the net flow across S :
∬S F ⋅ n dS , where n is the unit normal. This is computed by parametrizing S by
r(u, v) and integrating F ⋅ (ru × rv ) du dv . Flux integrals appear in physics (e.g.
​
​
​
Gauss’s law for electric fields, mass flow through a surface in fluid mechanics).
10. Green’s Theorem (2D)
Green’s Theorem connects a closed line integral in the plane to a double integral over the
region it encloses
tutorial.math.lam…
. If C is a positively oriented (counterclockwise) simple
closed curve bounding region D , and P (x, y), Q(x, y) have continuous partials on D ,
then
∮ P dx + Q dy = ∬ (
​
​
C
D
∂Q ∂P
−
) dA.
∂x
∂y
​
​
This is essentially the 2D version of Stokes’ theorem. It relates circulation of ⟨P , Q⟩
around C to the curl component ∂Q/∂x − ∂P /∂y over D .
Example: Let C be the unit circle x2 + y 2 = 1. Take P = −y , Q = x. Then
∮C P dx + Q dy equals ∬D (∂Q/∂x − ∂P /∂y) dA = ∬D (1 − (−1)) dA =
2 Area(D) = 2π . Indeed, directly parameterizing C by (cos t, sin t) yields
2π
∮C (− sin t)(− sin t) dt + (cos t)(cos t) dt = ∫0 dt = 2π .
​
​
​
​
​
Green’s Theorem is widely used to compute areas and line integrals. For instance, setting
P = −y/2, Q = x/2 in Green’s Theorem shows area(D) = ∮C x dy .
​
11. Divergence Theorem (Gauss’s Theorem)
In three dimensions, the Divergence Theorem generalizes Green’s Theorem. It relates the
flux of F out of a closed surface S to the volume integral of ∇ ⋅ F inside. Precisely, if S
is a closed surface bounding a solid region W , then
∬ F ⋅ n dS = ∭ (∇ ⋅ F) dV .
​
​
S
W
This means the total “outflow” of F through the boundary equals the sum of the sources
minus sinks inside. It underlies Gauss’s laws in physics (e.g. Gauss’s law for magnetism
states ∇ ⋅ B = 0 so the net magnetic flux is zero
math.libretexts.org
).
12. Stokes’ Theorem
Stokes’ Theorem unifies Green’s Theorem and the Divergence Theorem using the
language of surfaces: it states that for any smooth oriented surface S with boundary
curve ∂S ,
∮
​
∂S
F ⋅ dr = ∬ (∇ × F) ⋅ n dS.
​
S
Thus the circulation of F around the edge of S equals the flux of ∇ × F through S . In
the plane this reduces to Green’s Theorem, and over a closed surface it implies the
Divergence Theorem via the identity ∇ ⋅ (∇ × F) = 0. Stokes’ Theorem is
fundamental in electromagnetism and fluid dynamics (e.g. Faraday’s law ∮ E ⋅ dr =
− dtd ∬ B ⋅ dS).
​
13. Real-World Applications
Multivariable calculus is everywhere in science and engineering:
Physics: Modeling electromagnetic and gravitational fields (vector fields), fluid flow
and heat flow (via partial differential equations that use divergence and curl), and
classical mechanics (trajectories as vector-valued functions). Maxwell’s equations
and the Navier-Stokes equations are formulated in vector calculus form (div, curl).
Engineering: In gradient-based optimization (e.g. minimizing energy or cost
functions), in stress-strain analysis (tensors derived from derivatives), and in
computing volumes/masses of objects with varying density (multiple integrals).
Economics: Production and utility functions often depend on multiple inputs; partial
derivatives give marginal rates of substitution. Optimization with Lagrange
multipliers models constrained economic decisions.
Computer Graphics: Transforming 3D shapes uses multivariable geometry and
calculus (surface normals via gradients, parametrized surfaces, lighting via flux
integrals).
14. Differential Forms (Advanced Topic)
For a higher level of abstraction, differential forms unify many ideas. A differential k form is an object that can be integrated over k -dimensional surfaces. For example, a 1form looks like f (x, y) dx + g(x, y) dy (integrated along a curve) and a 2-form in 3D
looks like P dy ∧ dz + Q dz ∧ dx + R dx ∧ dy (integrated over a surface). The
exterior derivative d generalizes grad, curl, div: e.g. d(f dx) = fx dx ∧ dy , etc. Stokes’
​
theorem is compactly ∫∂Ω ω = ∫Ω dω for any form ω . In physics, forms streamline
​
​
Maxwell’s equations and simplify coordinate changes
en.wikipedia.org
. (For readers
interested in the geometric viewpoint, see texts on differential geometry or Calculus on
Manifolds by Spivak.)
Differential Forms Example: The expression f (x, y) dx + g(x, y) dy is a 1-form
whose integral over a path C gives a line integral ∫C f dx + g dy . Likewise P dy ∧
​
dz + Q dz ∧ dx + R dx ∧ dy is a 2-form in R3 whose integral over a surface gives
the flux ∬S P dy dz + Q dz dx + R dx dy . The operator d sends 1-forms to 2-forms,
​
encoding curl and divergence in algebraic form
15. Resources and Practice
en.wikipedia.org
.
Textbooks: Standard textbooks include Calculus by James Stewart (multivariable
edition), Vector Calculus by Marsden & Tromba, and Multivariable Calculus by Ron Larson.
For a free textbook, see Multivariable Calculus by Don Shimamoto (Swarthmore)
open.umn.edu
, available via Open Textbook Library. For a more theoretical approach, Vector
Calculus, Linear Algebra and Differential Forms: A Unified Approach by Hubbard & Hubbard
(Springer) is excellent (available online). Michael Spivak’s Calculus on Manifolds covers
forms rigorously (but is challenging). Lecture notes like Paul’s Online Math Notes
(Calculus III section) provide concise explanations and many worked examples.
Online Courses and Videos: MIT OpenCourseWare (course 18.02) has video lectures,
notes and problem sets
ocw.mit.edu
. Khan Academy offers free video tutorials on
multivariable calculus. YouTube channels like 3Blue1Brown and Professor Leonard have
visual/intuitive multivariable calculus series. Coursera or edX have courses from
institutions like MIT, University of Michigan, or Imperial College. For practice, find
problem sets in textbooks or on OCW sites. Paul’s Online also has practice problems and
solutions.
Practice: Solve many example problems – compute partial derivatives and gradients of
given functions, tangent planes, directional derivatives; evaluate double/triple integrals in
Cartesian and polar/spherical; compute line/surface integrals for simple fields; verify
Green’s/Stokes’ Theorem on test cases; solve optimization problems with and without
constraints. Practice visualizing vector fields and computing divergence/curl on simple
fields. Typical problems appear in Stewart’s textbook and in MIT 18.02 problem sets
(available on OCW)
ocw.mit.edu
.
Advanced Topics (optional): Once core topics are mastered, one can explore further:
multivariable Fourier series and PDEs, differential geometry (curvature of surfaces,
Christoffel symbols), tensors, or numerical methods for multiple integrals and
optimization. But mastering the fundamentals above provides a strong foundation for
applications in science and engineering.
Summary: Multivariable calculus builds on single-variable ideas but introduces new
geometry and algebra (gradients, multiple integrals, vector operators). It is the language
of fields and flows. By studying vectors, partial derivatives, multiple integration, and the
major theorems (Green/Stokes/Divergence), a student gains powerful tools for modeling
anything from fluid flow to electromagnetic fields. Consistent practice and visualization
(using graphs and diagrams) solidify understanding. The resources above, along with
problem-solving, will guide a motivated learner through all the core topics and prepare
them for applications in physics, engineering, and beyond.
Recommended Resources (condensed):
Textbooks: Stewart Multivariable Calculus; Marsden & Tromba Vector Calculus;
Multivariable Calculus by Shimamoto
open.umn.edu
(free); Hubbard & Hubbard Vector
Calc with Differential Forms.
Online Courses: MIT OCW 18.02 (Auroux)
ocw.mit.edu
; Khan Academy; Coursera/edX
courses (e.g. from Imperial College).
Videos: MIT 18.02 lectures; 3Blue1Brown calculus series; Professor Leonard’s
calculus playlist.
Notes/Practice: Paul’s Online Notes (Calculus III) and exercises; APEX Calculus
(free text) exercises; MIT OCW problem sets (with solutions)
ocw.mit.edu
.
Citations
Gradient - Wikipedia
https://en.wikipedia.org/wiki/Gradient
Gradient - Wikipedia
https://en.wikipedia.org/wiki/Gradient
Gradient - Wikipedia
https://en.wikipedia.org/wiki/Gradient
Gradient - Wikipedia
https://en.wikipedia.org/wiki/Gradient
Introduction to Taylor's theorem for multivariable functions - Math Insight
https://mathinsight.org/taylors_theorem_multivariable_introduction
15.5: Divergence and Curl - Mathematics LibreTexts
https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_1
15.5: Divergence and Curl - Mathematics LibreTexts
https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_1
15.5: Divergence and Curl - Mathematics LibreTexts
https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_1
Curl (mathematics) - Wikipedia
https://en.wikipedia.org/wiki/Curl_(mathematics)
Gradient - Wikipedia
https://en.wikipedia.org/wiki/Gradient
Calculus III - Green's Theorem
https://tutorial.math.lamar.edu/classes/calciii/GreensTheorem.aspx
16.5: Divergence and Curl - Mathematics LibreTexts
https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/16%3A_Vector_Calculus/16
Differential form - Wikipedia
https://en.wikipedia.org/wiki/Differential_form
Multivariable Calculus - Open Textbook Library
https://open.umn.edu/opentextbooks/textbooks/780
Multivariable Calculus | Mathematics | MIT OpenCourseWare
https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/
Multivariable Calculus | Mathematics | MIT OpenCourseWare
https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/
All Sources
en.wikipedia
ocw.mit
mathinsight
math.libretexts
tutorial.math.lamar
open.umn