Difficulty: Intermediate | Prerequisites: Calculus I and II (derivatives, integrals, multivariable functions)
Calculus III extends single-variable calculus into multiple dimensions. Where Calculus I and II dealt with functions of one variable, this course works with vector fields and functions of two or three variables. The topics here, gradient, divergence, curl, Green's Theorem, the Divergence Theorem and Taylor's Theorem, form the backbone of the course and appear across physics, engineering and applied mathematics. If you are comfortable with partial derivatives and basic integration, you have what you need to start.
Calculus III extends the derivative and integral toolkit into multiple dimensions. You will work with vector fields (gradient, divergence, curl), learn theorems that convert hard integrals into easier ones (Green's Theorem, the Divergence Theorem), and use Taylor's Theorem to approximate complicated functions with polynomials. Master these and you have the mathematical backbone for physics, fluid dynamics and engineering.
Gradient (∇f)
A vector that collects all partial derivatives of a scalar function f. It points in the direction of steepest increase at any given point, and its magnitude tells you how steep that increase is.
In simple terms, the gradient is a signpost that says "this way up" on a surface, and how long the arrow is tells you how sharply the surface rises.
Divergence (∇ · F)
A scalar quantity that measures how much a vector field spreads out from (or converges into) a point. Positive divergence means net outward flow; negative means net inward flow; zero means the field neither expands nor compresses locally.
Think of it as standing at a point in a flow of water: divergence tells you whether the water is being pushed away from you (a source) or pulled toward you (a sink).
Curl (∇ × F)
A vector that measures the local rotation, or "swirl," of a vector field at a point. Its direction is the axis of rotation (by the right-hand rule) and its magnitude is the rate of rotation.
Think of it as dropping a tiny paddlewheel into a flowing river: the curl tells you which way and how fast the wheel spins.
Vector field
A function that assigns a vector to every point in a region of space. Wind maps and magnetic field diagrams are common examples.
Line integral
An integral evaluated along a curve rather than over an interval. Used to compute work done by a force along a path, or circulation of a field around a loop.
Flux
The total amount of a vector field passing through a surface. Net outward flux through a closed surface relates to the divergence inside (that is the Divergence Theorem).
Green's Theorem
A result that converts a line integral around a simple closed curve in the plane into a double integral over the region it encloses (and vice versa). It links circulation and curl in two dimensions.
In simple terms, instead of walking the boundary and adding up a field's contributions, you can sweep across the interior. Whichever integral is easier, use that one.
Divergence Theorem (Gauss's Theorem)
Generalises Green's Theorem to three dimensions: the total outward flux of a vector field through a closed surface equals the triple integral of the divergence over the enclosed volume.
In simple terms, if you want to know how much "stuff" is leaving a region, you can either measure it at every point on the boundary or add up all the sources and sinks inside.
Taylor's Theorem
States that a sufficiently smooth function can be approximated near a point by a polynomial whose coefficients come from the function's derivatives at that point. The remainder term bounds the approximation error.
In simple terms, you replace a complicated function with a polynomial that matches its value, slope, curvature (and so on) at one point, then use that polynomial for calculations nearby.
For a scalar function f(x, y, z), the gradient is ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z).
The gradient always points in the direction of greatest increase of f.
Its magnitude |∇f| gives the rate of that increase.
At a local maximum or minimum, ∇f = 0 (the zero vector).
The gradient is perpendicular to the level curves (2D) or level surfaces (3D) of f.
Example: For f(x, y) = x² + y², the gradient is ∇f = (2x, 2y). At the point (1, 1) the gradient is (2, 2), pointing away from the origin, confirming the function increases as you move outward.
For a vector field F = (P, Q, R), the divergence is ∇ · F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.
Divergence is a scalar, not a vector.
Positive divergence at a point means the field acts as a source there; negative means it acts as a sink.
A field with zero divergence everywhere is called incompressible or solenoidal (e.g. an incompressible fluid flow).
Example: For F = (x, y, z), the divergence is ∇ · F = 1 + 1 + 1 = 3. The field expands uniformly in every direction.
For a vector field F = (P, Q, R), the curl is ∇ × F = (∂R/∂y − ∂Q/∂z, ∂P/∂z − ∂R/∂x, ∂Q/∂x − ∂P/∂y).
Curl is a vector, not a scalar.
A field with zero curl everywhere is called irrotational (or conservative in simply connected regions).
If F is conservative (i.e. F = ∇f for some scalar f), then ∇ × F = 0.
Example: For F = (−y, x, 0), the curl is ∇ × F = (0, 0, 2). The field rotates counterclockwise around the z-axis.
The gradient takes a scalar field to a vector field.
Divergence takes a vector field to a scalar field.
Curl takes a vector field to another vector field.
Two identities tie them together:
∇ × (∇f) = 0 for any smooth scalar f (the curl of a gradient is always zero).
∇ · (∇ × F) = 0 for any smooth vector field F (the divergence of a curl is always zero).
Applies to a vector field F = (P, Q) in two dimensions and a simple, closed, positively oriented (counterclockwise) curve C enclosing a region D.
Statement: ∮_C (P dx + Q dy) = ∬_D (∂Q/∂x − ∂P/∂y) dA.
The left side is a line integral around the boundary; the right side is a double integral over the interior.
Use Green's Theorem when one side of the equation is easier to evaluate than the other.
Special case for area: the area of D equals ½ ∮_C (x dy − y dx).
Real-world application: Green's Theorem is used in computational geometry to calculate the area of irregular shapes from their boundary coordinates, which is how GIS software measures land parcels.
Extends Green's Theorem into three dimensions.
Applies to a vector field F in ℝ³ and a closed surface S enclosing a volume V.
Statement: ∬_S F · dS = ∭_V (∇ · F) dV.
The left side is the total outward flux through the surface; the right side is the integral of the divergence over the volume.
Useful when computing a surface integral directly is difficult but the divergence inside is simple (or vice versa).
Real-world application: Engineers use the Divergence Theorem in electrostatics (Gauss's Law) to compute electric fields around charged objects by converting a surface integral into a volume integral over the charge distribution.
For a function f that is (n + 1) times differentiable near a point a, the nth-degree Taylor polynomial is:
T_n(x) = f(a) + f'(a)(x − a) + f''(a)(x − a)²/2! + … + f⁽ⁿ⁾(a)(x − a)ⁿ/n!
The remainder (error) term R_n(x) satisfies |R_n(x)| ≤ M|x − a|ⁿ⁺¹/(n + 1)! where M bounds |f⁽ⁿ⁺¹⁾| on the interval.
When a = 0, the expansion is called a Maclaurin series.
Common Maclaurin series worth memorising:
eˣ = 1 + x + x²/2! + x³/3! + …
sin x = x − x³/3! + x⁵/5! − …
cos x = 1 − x²/2! + x⁴/4! − …
1/(1 − x) = 1 + x + x² + x³ + … (for |x| < 1)
In Calculus III, Taylor's Theorem extends to multivariable functions. The second-order Taylor expansion of f(x, y) near (a, b) involves partial derivatives f_x, f_y, f_xx, f_xy and f_yy.
Real-world application: Taylor approximations underpin how calculators evaluate trigonometric and exponential functions: they compute a polynomial that is accurate to many decimal places near the expansion point.
Students often think divergence and curl are inverses or negatives of each other. They are not. Divergence is a scalar (how much the field spreads); curl is a vector (how much the field rotates). They measure entirely different properties.
Students sometimes assume that if a vector field "looks circular," its divergence must be zero. Divergence depends on whether the field expands or compresses, not on its shape. A circular field can still have nonzero divergence.
A common error is applying Green's Theorem to a curve that is not closed or not simple (i.e. it crosses itself). The theorem requires a simple, closed, positively oriented boundary.
Students frequently confuse the direction of the curl vector with the direction of flow. The curl vector points along the axis of rotation (via the right-hand rule), not in the direction the fluid moves.
⚠️ Know the exact formulas for gradient, divergence and curl. Exams frequently ask you to compute these for a given field.
⚠️ Be able to state Green's Theorem and the Divergence Theorem precisely, including all conditions (closed curve, simple region, outward normal).
⚠️ Expect at least one problem that asks you to choose whether to evaluate the line/surface integral directly or convert via a theorem. Picking the easier side is the skill being tested.
⚠️ Taylor's Theorem questions often ask you to write the nth-degree polynomial for a given function and bound the error. Memorise the remainder formula.
⚠️ The two vector identities, curl(grad f) = 0 and div(curl F) = 0, appear in true/false and proof questions regularly.
True or False: The gradient of a scalar function is a scalar.
False. The gradient is a vector.
True or False: If ∇ × F = 0 everywhere in a simply connected region, then F is conservative.
True.
Fill in the blank: Green's Theorem relates a ______ integral around a closed curve to a ______ integral over the enclosed region.
line; double
True or False: The divergence of a curl is always zero.
True. ∇ · (∇ × F) = 0 for any smooth F.
Fill in the blank: A Maclaurin series is a Taylor series expanded about a = ______.
0
Q: Compute the gradient of f(x, y, z) = x²y + yz³. At what point(s) is each component of the gradient zero?
A: ∇f = (2xy, x² + z³, 3yz²). Setting each component to zero: 2xy = 0, x² + z³ = 0, 3yz² = 0. The origin (0, 0, 0) satisfies all three.
Q: Let F = (y², −x², 0). Compute ∇ · F and ∇ × F.
A: ∇ · F = ∂(y²)/∂x + ∂(−x²)/∂y + ∂(0)/∂z = 0 + 0 + 0 = 0. ∇ × F = (∂(0)/∂y − ∂(−x²)/∂z, ∂(y²)/∂z − ∂(0)/∂x, ∂(−x²)/∂x − ∂(y²)/∂y) = (0, 0, −2x − 2y).
Q: State Green's Theorem. Under what conditions does it apply?
A: ∮_C (P dx + Q dy) = ∬_D (∂Q/∂x − ∂P/∂y) dA. It applies when C is a simple, closed, positively oriented (counterclockwise) curve bounding a region D, and P and Q have continuous partial derivatives on an open region containing D.
Q: Use the Divergence Theorem to evaluate ∬_S F · dS where F = (x, y, z) and S is the unit sphere.
A: ∇ · F = 1 + 1 + 1 = 3. By the Divergence Theorem, ∬_S F · dS = ∭_V 3 dV = 3 × (4π/3) = 4π.
Q: Write the third-degree Taylor polynomial for eˣ centred at a = 0.
A: T₃(x) = 1 + x + x²/2 + x³/6.
Green's Theorem and the Divergence Theorem are both special cases of Stokes' Theorem, the most general result in vector calculus linking integrals over a boundary to integrals over the region it bounds. Expect to meet the full version of Stokes' Theorem later in the course.
Gradient, divergence and curl reappear throughout physics: Maxwell's equations in electromagnetism are stated entirely in these terms, and fluid dynamics relies on the same operators to describe flow, vorticity and compressibility.
Taylor's Theorem connects forward to power series representations of functions and to the study of convergence, which you will use when solving differential equations by series methods.
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