Glossary

The cross-partials check: \mathbf{F}=(P,Q) is conservative on a simply connected region exactly when P_y=Q_x everywhere.

A vector field in space is conservative on a simply connected region exactly when its curl is zero everywhere.

The single highest (or lowest) value f takes over its entire domain, not just near one point.

A moving particle's velocity derivative, \mathbf{a}(t)=\mathbf{v}'(t)=\mathbf{r}''(t).

The length of a curve traced by \mathbf{r}(t), found by integrating speed over time.

Re-timing a curve so its own parameter measures distance traveled instead of time.

The two-variable generalization of a function's average, f_{\text{avg}}=\dfrac{1}{A(R)}\iint_R f\,dA.

The third vector of the TNB frame, \mathbf{B}=\mathbf{T}\times\mathbf{N}, perpendicular to both \mathbf{T} and \mathbf{N}.

The Fundamental Theorem for Line Integrals, both forms of Green's Theorem, Stokes' Theorem, and the Divergence Theorem are one statement, made at four different dimensions.

Substituting new coordinates into a double or triple integral, trading a hard region or integrand for an easy one at the cost of the Jacobian.

How much a vector field pushes along a closed curve, totaled all the way around — the paddle-wheel reading summed over an entire loop.

If a function's mixed partials are continuous, the order of differentiation doesn't matter — f_{xy}=f_{yx}.

A domain with no missing boundary points and no infinite extent — the condition that guarantees a continuous function has an absolute max and min.

A vector field that is someone's gradient — its line integral between two points never depends on the path taken.

Finding the extrema of an objective function f(x,y) restricted to a constraint set g(x,y)=k.

A fixed relationship tying a problem's variables together, used to eliminate one variable by substitution.

f is continuous at (a,b) when f(a,b) is defined, the limit there exists, and the two agree.

A vector-valued function is continuous exactly where each of its three component functions is.

A whole family of a function's level curves, drawn together at evenly spaced output values.

A point where every partial derivative is zero (or fails to exist) — the two-variable analogue of f'(x)=0.

A way to multiply two vectors that returns a new vector perpendicular to both.

A vector measuring whether a field is spinning around a point, pointing along the axis of fastest rotation with length equal to the spin rate.

How fast the unit tangent vector turns per unit of arc length, \kappa=|\mathbf{r}'\times\mathbf{r}''|/|\mathbf{r}'|^3.

A surface built by sliding a fixed 2D curve straight along an axis whose variable is missing from the equation.

Polar coordinates (r,\theta) in the xy-plane left unchanged, with z added on; volume element dV=r\,dz\,dr\,d\theta.

Mass per unit area at a point, \rho(x,y), whose integral over a region gives total mass.

The tangent vector \mathbf{r}'(t), found by differentiating each component separately.

The stronger condition, beyond both partials existing, that a linear plane approximates f well from EVERY direction at a point.

The rate of change of f in an arbitrary direction, specified by a unit vector, generalizing the partial derivatives.

How far a point sits from a line in space, found from the cross product: D=|\overrightarrow{P_0P_1}\times\mathbf{v}|/|\mathbf{v}|.

How far a point sits from a plane, found by projecting onto the plane's unit normal.

A scalar measuring whether a vector field is, on balance, spreading material away from a point or pulling it in.

Trades a closed surface's total flux for a triple integral of divergence over the solid region it encloses.

The region of the xy-plane (or space) where a multivariable function's formula produces a defined output.

A way to multiply two vectors that returns a number measuring how much they agree in direction.

The limit of a double Riemann sum \sum f(x^*,y^*)\,\Delta A over a region — signed volume under a surface z=f(x,y).

Converting \iint_R f\,dA into polar bounds with the extra factor dA=r\,dr\,d\theta.

How much a vector field crosses through a closed curve, outward or inward — the net amount of "stuff" the field carries across the boundary.

The 3D surface analogue of flux in the plane, \iint_S\mathbf{F}\cdot d\mathbf{S}, measuring how much a vector field flows through an oriented surface.

A theorem guaranteeing a double integral over a rectangle equals either iterated order, and both orders give the same number.

A rule z=f(x,y) (or w=f(x,y,z)) that assigns exactly one output to each ordered pair or triple of inputs.

For a conservative field, a line integral collapses to a two-number subtraction: \int_C \nabla f\cdot d\mathbf{r} = f(B)-f(A), depending only on endpoints.

A solid squeezed between two surfaces over a shadow region in one of the coordinate planes.

An algorithm that repeatedly steps opposite the gradient to hunt for a local minimum.

The vector field built by taking the gradient of a scalar function — an arrow at every point pointing in the direction that function increases fastest.

\nabla f always points exactly perpendicular to the level curve passing through that point.

The vector \nabla f=\langle f_x,f_y\rangle of a function's partial derivatives, whose dot product with a unit vector gives the directional derivative in that direction.

The set of points (x,y,f(x,y)) in space — a surface, one dimension bigger than a single-variable graph's curve.

Trades a circulation integral around a closed curve for a double integral of curl over the region it encloses.

Green's Theorem's two versions: the tangential form trades circulation for a curl integral, the normal form trades flux for a divergence integral.

The matrix of second partial derivatives, \begin{pmatrix} f_{xx} & f_{xy} \\ f_{xy} & f_{yy}\end{pmatrix}, whose determinant is the second derivative test's discriminant D.

Using the multivariable chain rule on F(x,y)=0 gives dy/dx=-F_x/F_y directly, without solving for y.

The property that a vector field's line integral between two points gives the same answer no matter which curve connects them — equivalent to being conservative.

A double integral computed as two nested single-variable integrals, one variable integrated at a time.

A determinant of partial derivatives measuring how much a coordinate change stretches or shrinks area (or volume) at a point.

A method for constrained extrema using \nabla f=\lambda\nabla g — the tangency of the objective's and constraint's gradients.

The three-variable extension \nabla f=\lambda\nabla g+\mu\nabla h, used when two constraint surfaces intersect in a curve.

The line minimizing the sum of squared errors E(m,b)=\sum_i(y_i-(mx_i+b))^2 — an unconstrained minimum in disguise.

The set of points where f(x,y) equals one fixed output value k, drawn flat in the xy-plane.

The set of points where a three-variable function f(x,y,z) equals a fixed output — a surface floating in space.

The value f(x,y) approaches as (x,y) approaches a point, required to agree along every possible path in.

A point plus a direction vector, swept by a parameter: \mathbf{r}(t)=\mathbf{r}_0+t\mathbf{v}.

A curve-analogue of a definite integral, \int_C f\,ds, that weights a curve by a scalar function and adds up, using arc length ds.

The integral \int_C \mathbf{F}\cdot d\mathbf{r}, adding up the component of a vector field along a curve's direction of travel — physically, work.

Using a surface's tangent plane, instead of the true function, to cheaply estimate nearby values.

A local max or min — a point that beats only its immediate neighbors, not necessarily the whole domain.

A vector's length, |\mathbf{v}|=\sqrt{v_1^2+v_2^2+v_3^2} — the distance formula from the origin.

The total mass of a flat plate with variable density, m=\iint_R\rho(x,y)\,dA.

A second-order partial derivative taken with respect to two different variables, such as f_{xy}.

Mass weighted by the square of its distance from an axis, I=\iint_R r^2\rho\,dA — a measure of resistance to rotation.

M_y and M_x weight mass by its distance from an axis; (\bar x,\bar y)=(M_y/m,\ M_x/m) is the plate's balance point.

If z=f(x,y) and x,y both depend on t, then dz/dt sums both partials' contributions: f_x\,dx/dt+f_y\,dy/dt.

The linear system from setting \partial E/\partial m=0 and \partial E/\partial b=0, solved for a regression line's slope and intercept.

The gradient \nabla F of an implicit surface F(x,y,z)=k is a ready-made normal vector, letting the tangent plane be written down without solving for z.

A vector pointing straight off a plane (or surface), perpendicular to everything lying in it.

The quantity being optimized in a word problem, written as a function of the problem's variables.

A consistent choice of which side of a surface counts as "outward," made everywhere at once — not every surface admits one.

The slanted 3D brick shape swept out by three vectors that don't all lie in one plane.

A surface swept out by two independent parameters, \mathbf{r}(u,v) = \langle x(u,v),\,y(u,v),\,z(u,v)\rangle — free of the one-height-per-point restriction of a graph.

The rate of change of a multivariable function in one variable's direction, with every other variable held fixed.

A flat 2D sheet in space, pinned down by one point on it plus a normal vector.

A scalar function f whose gradient reproduces a given conservative vector field, \nabla f = \mathbf{F}.

The unit vector \mathbf{N} pointing the direction \mathbf{T} is turning toward.

One of six 3D surface families — the 3D cousins of the conic sections — from a squared-variable equation.

The set of z-values a multivariable function's surface actually reaches.

A critical point that curves upward in one direction and downward in another, with no single-variable analogue.

The signed length of one vector's shadow along another, \text{comp}_{\mathbf{u}}\mathbf{v}=\dfrac{\mathbf{v}\cdot\mathbf{u}}{|\mathbf{u}|}.

The box product \mathbf{u}\cdot(\mathbf{v}\times\mathbf{w}), whose absolute value is a parallelepiped's volume.

The two-variable test using D=f_{xx}f_{yy}-f_{xy}^2 to classify a critical point as a local max, local min, saddle, or inconclusive.

The 3!=6 ways to order dx, dy, dz in a triple integral — all give the same value, but differ in algebraic difficulty.

The magnitude of velocity, |\mathbf{r}'(t)| — a positive number with no direction attached.

Distance \rho from the origin plus two angles \theta,\phi; volume element dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta.

The gradient's direction is the direction f increases fastest, and its length is that fastest rate.

Lifts Green's tangential form off the flat plane: a curve's circulation traded for curl integrated as flux through any curved surface it bounds.

The area of a curved surface, S=\iint_R\sqrt{1+f_x^2+f_y^2}\,dA, built by summing tiny tangent-plane patches.

The true curved area of a parametric surface, found by weighting the flat parameter domain by the stretch factor |\mathbf{r}_u\times\mathbf{r}_v|.

A surface integral \iint_S f\,dS that weights each patch of a curved surface by a varying scalar density before summing.

A surface swept out by spinning a profile curve around an axis, using the rotation angle as one parameter and position along the curve as the other.

Since Stokes' Theorem's curl-flux only depends on the boundary curve, any two oriented surfaces sharing that same boundary give the identical integral.

The flat plane through a point on a surface that matches the surface's height and both partial-derivative slopes there.

The flat plane that best approximates a parametric surface at a point, spanned by the two partial-derivative vectors \mathbf{r}_u and \mathbf{r}_v.

Acceleration split into a part along velocity (a_T, speeding up/slowing down) and a part perpendicular to it (a_N, turning).

Adding the three second-order terms (f_{xx}, f_{xy}, f_{yy}) to the tangent plane produces a paraboloid that also matches the surface's curvature.

The moving frame of three mutually perpendicular unit vectors \mathbf{T}, \mathbf{N}, \mathbf{B} that rides along a curve.

How fast a curve's osculating plane tilts as it twists out of that plane, per unit arc length.

The differential-notation version of linear approximation: dz=f_x\,dx+f_y\,dy.

The cross-section left when a surface is sliced by a plane that freezes one input variable at a constant.

\iiint_E f\,dV — the limit of a Riemann sum over small boxes filling a solid, one dimension up from a double integral.

To prove a two-variable limit does not exist, find two different paths into the point where f settles on two different values.

A plane region a\le x\le b, g_1(x)\le y\le g_2(x) — bounded left/right by constants, top/bottom by functions of x.

A plane region bounded top/bottom by constants and left/right by functions of y, integrated dx first.

The direction \mathbf{r}'(t) points, with speed divided out: \mathbf{T}(t)=\mathbf{r}'(t)/|\mathbf{r}'(t)|.

A vector of magnitude exactly 1, used to isolate direction alone.

A magnitude and a direction bundled into one object, written \langle x,y,z\rangle or \mathbf{v}.

The two basic vector operations — combine two vectors componentwise, or stretch/shrink/reverse one by a number.

A function that assigns an arrow — a vector with its own length and direction — to every point in a region, instead of a single number.

The shadow one vector casts onto another, as a full vector pointing along the second one.

A function assigning a vector to every input t: \mathbf{r}(t)=\langle f(t),g(t),h(t)\rangle.

A moving particle's position derivative, \mathbf{v}(t)=\mathbf{r}'(t) — a vector with both speed and direction.

The dot product of a constant force and a displacement, W=\mathbf{F}\cdot\mathbf{d}.