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Every rule and concept in Multivariable Calculus, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (77)

A line is a point and a direction: The line through $P_0 = (x_0, y_0, z_0)$ with direction $\mathbf v = (a, b, c)$ is $\mathbf r(t) = P_0 + t\mathbf v$ , that is $x = x_0 + at$ , $y = y_0 + bt$ , $z = z_0 + ct$ .; Lines & Planes in Space › Lines in Space and How Two Lines Meet
A plane is a point and a normal: Every plane in space is captured by ONE linear equation $ax + by + cz = d$ , and the coefficient vector $\mathbf n = (a, b, c)$ is no accident: it is the NORMAL, perpendicular to the plane.; Lines & Planes in Space › Equations of Lines and Planes
Acceleration has a speeding-up part and a turning part: With speed $v = |\mathbf v|$ , \[ a = $\frac{dv}{dt}$ T + v^2 N,\] where $\mathbf N$ is the unit normal pointing toward the inside of the turn.; Curves & Motion › Tangential and Normal Acceleration
All distances are projections onto the normal: Distance from a point $P_0 = (x_0, y_0, z_0)$ to the plane $ax + by + cz = d$ : $\dfrac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}}$ .; Lines & Planes in Space › Equations of Lines and Planes
Check the inside, the edges and the corners: A continuous $f$ on a closed, bounded region attains an absolute maximum and minimum.; Optimization & Lagrange Multipliers › Absolute Extrema on a Closed Region
Choose the easier side: $\displaystyle\oint_C P\,dx + Q\,dy = \iint_R (Q_x - P_y)\,dA$ when $C$ is the boundary of $R$ traversed counterclockwise, with the region on the left.; Vector Calculus › Green's Theorem in Practice
Choosing coordinates: read the region, then the integrand: The decision tree: does the region have an axis of circular symmetry?; Triple Integrals & Coordinate Systems › Going Deeper: Explanations & Worked Problems
Columns or rows: A region between two graphs, $g_1(x) \le y \le g_2(x)$ for $a \le x \le b$ , is integrated in columns: $\displaystyle\int_a^b\!\int_{g_1(x)}^{g_2(x)} f\,dy\,dx$ .; Double Integrals › Regions Between Curves, and Reversing the Order
Conservative fields have potentials: If $\mathbf F = \nabla f$ then $\int_C \mathbf F\cdot d\mathbf r = f(\text{end}) - f(\text{start})$ — the fundamental theorem for line integrals: path-independent, and zero around every loop.; Vector Calculus › Fields, Circulation, and Flux
Conservative or not: the full workflow: Given $\mathbf F = (P, Q)$ , the question "does a potential exist?" has a mechanical answer.; Vector Calculus › Going Deeper: Explanations & Worked Problems
Critical points and the D-test: Interior extrema need $\nabla f = \mathbf 0$ .; Optimization & Lagrange Multipliers › Free and Constrained Optimization
Cross product = area machine: $\mathbf u\times\mathbf v$ is perpendicular to both inputs (right-hand rule) with magnitude $|\mathbf u||\mathbf v|\sin\theta$ — the AREA of the parallelogram they span.; Vectors in Space › Dot and Cross Products
Curvature is turning per unit length: The unit tangent is $\mathbf T = \dfrac{\mathbf r'}{|\mathbf r'|}$ .; Curves & Motion › Curvature from Any Parametrization
Curvature measures how hard you turn: For a graph $y = f(x)$ : $\kappa = \dfrac{|y''|}{(1 + y'^2)^{3/2}}$ .; Curves & Motion › Vector-Valued Functions
Differentiate coordinate by coordinate: A path $\mathbf r(t) = (x(t), y(t), z(t))$ has velocity $\mathbf r'(t)$ (tangent to the path) and speed $|\mathbf r'(t)|$ .; Curves & Motion › Vector-Valued Functions
Divergence and curl, in one breath: $\nabla\cdot\mathbf F$ measures how much the field SPREADS from a point (source strength); $\nabla\times\mathbf F$ measures how much it ROTATES.; Vector Calculus › Fields, Circulation, and Flux
Dot product = angle detector: $\mathbf u\cdot\mathbf v = |\mathbf u||\mathbf v|\cos\theta$ , computed coordinate-wise as $u_1v_1 + u_2v_2 + u_3v_3$ .; Vectors in Space › Dot and Cross Products
Flux: how much field crosses a surface: The surface integral $\iint_S \mathbf F\cdot d\mathbf S = \iint_S \mathbf F\cdot\mathbf n\,dS$ measures net flow through $S$ : only the component of $\mathbf F$ along the unit normal $\mathbf n$ counts.; Surface Integrals, Stokes & Divergence › The Capstone Theorems
Freeze everything but one variable: For $f(x, y)$ , the partial $f_x$ differentiates in $x$ while treating $y$ as a CONSTANT (and vice versa for $f_y$ ).; Partial Derivatives & the Gradient › Partial Derivatives
Green's theorem trades a loop for a region: $\oint_C P\,dx + Q\,dy = \iint_R (Q_x - P_y)\,dA$ for a counterclockwise loop.; Vector Calculus › Fields, Circulation, and Flux
Implicit differentiation, industrialized: A curve given implicitly by $F(x, y) = 0$ has slope $\dfrac{dy}{dx} = -\dfrac{F_x}{F_y}$ wherever $F_y \ne 0$ — no solving for $y$ , ever.; The Chain Rule & Implicit Differentiation › Differentiating Through Dependencies
Implicit partial derivatives: If $F(x, y, z) = 0$ defines $z$ as a function of $x$ and $y$ near a point where $F_z \ne 0$ , then \[ $\frac{\partial z}{\partial x}$ = - $\frac{F_x}{F_z}$ , $\frac{\partial z}{\partial y}$ = - $\frac{F_y}{F_z}$ .\] For a curve $F(x, y) = 0$ , $ $\dfrac{dy}{dx}$ = -…; The Chain Rule & Implicit Differentiation › Implicit Differentiation in Practice
Integrate one variable at a time: $\iint_R f\,dA$ over a rectangle is two ordinary integrals nested: inner in $y$ (holding $x$ ), then outer in $x$ — and Fubini says the order does not matter for continuous $f$ .; Double Integrals › Iterated Integration
Lagrange: gradients align at constrained extrema: To optimize $f$ subject to $g = c$ , solve $\nabla f = \lambda\nabla g$ : at the best point the level curve of $f$ is TANGENT to the constraint, so the gradients are parallel.; Optimization & Lagrange Multipliers › Free and Constrained Optimization
Let the boundary choose u and v: When a region is bounded by two pairs of curves $u(x, y) =$ const and $v(x, y) =$ const, take those expressions as the new variables: the region becomes a rectangle in the $uv$ -plane.; Change of Variables & the Jacobian › Choosing a Substitution
Limits from the inside out: Write $\displaystyle\iiint_E f\,dV = \int_a^b\!\int_{g_1(x)}^{g_2(x)}\!\int_{h_1(x, y)}^{h_2(x, y)} f\,dz\,dy\,dx$ .; Triple Integrals & Coordinate Systems › Setting Up Triple Integrals in Cartesian Coordinates
Normals do the work: The angle between two planes is the angle between their normals: $\cos\theta = \dfrac{|\mathbf n_1\cdot\mathbf n_2|}{|\mathbf n_1||\mathbf n_2|}$ .; Lines & Planes in Space › Angles, Intersections, and Distances
Parametrize, substitute, integrate: For a curve $C$ given by $\mathbf r(t)$ , $a \le t \le b$ : \[∫_C F· d r = ∫_a^b F( r(t))· r'(t) dt, ∫_C f ds = ∫_a^b f( r(t)) | r'(t)| dt.\] The first (work, circulation) changes…; Vector Calculus › Line Integrals: Work Along a Path
Partial derivatives from the limit up: The definition mirrors one-variable calculus exactly: $f_x(a,b) = \lim_{h\to0}\dfrac{f(a+h, b) - f(a, b)}{h}$ — wiggle $x$ only, watch the output, divide, take the limit.; Partial Derivatives & the Gradient › Going Deeper: Explanations & Worked Problems
Pick the side that is easier: Divergence theorem: $\displaystyle\oiint_S\mathbf F\cdot d\mathbf S = \iiint_E\nabla\cdot\mathbf F\,dV$ for the outward normal on the closed boundary $S$ of $E$ .; Surface Integrals, Stokes & Divergence › Using the Divergence and Stokes Theorems
Polar coordinates carry an extra r: On disks and rings, switch to $x = r\cos\theta$ , $y = r\sin\theta$ — but the area element becomes $dA = r\,dr\,d\theta$ .; Double Integrals › Iterated Integration
Projection: the shadow and the perpendicular remainder: The vector labelled $\mathrm{proj}_{\mathbf v}\mathbf u$ is the shadow of $\mathbf u$ on the line through $\mathbf v$ — the projection $\mathrm{proj}_{\mathbf v}\mathbf u = \dfrac{\mathbf u\cdot\mathbf v}{\mathbf v\cdot\mathbf v}\mathbf v$ .; Vectors in Space › Seeing It: Vectors in 2D & 3D
Proof sketch: Green's theorem on a rectangle: On $[a,b]\times[c,d]$ : $\iint Q_x\,dA = \int_c^d\left[Q(b,y) - Q(a,y)\right]dy$ , which is the line integral of $Q\,dy$ up the right side and down the left.; Vector Calculus › Proofs & Why It Matters
Proof sketch: the divergence theorem on a box: On a box, split the flux into three pairs of opposite faces.; Surface Integrals, Stokes & Divergence › Proofs & Why It Matters
Proof sketch: the second-derivative D-test: Near a critical point, Taylor gives $f \approx f(a,b) + \tfrac12\left(f_{xx}h^2 + 2f_{xy}hk + f_{yy}k^2\right)$ — a quadratic form $Q(h,k)$ .; Optimization & Lagrange Multipliers › Proofs & Why It Matters
Proof: |u×v| is the parallelogram area: The parallelogram on $\mathbf u, \mathbf v$ has base $|\mathbf u|$ and height $|\mathbf v|\sin\theta$ , so its area is $|\mathbf u||\mathbf v|\sin\theta$ .; Vectors in Space › Proofs & Why It Matters
Proof: arc length = ∫|r′(t)| dt: Chop $[a, b]$ into small steps $\Delta t$ .; Curves & Motion › Proofs & Why It Matters
Proof: Cauchy–Schwarz |u·v| ≤ |u||v|: For every real $t$ , $0 \le |\mathbf u + t\mathbf v|^2 = |\mathbf u|^2 + 2t(\mathbf u\cdot\mathbf v) + t^2|\mathbf v|^2$ — a quadratic in $t$ that is never negative, so its discriminant is $\le 0$ : $4( u· v)^2 - 4| u|^2| v|^2 …; Vectors in Space › Proofs & Why It Matters
Proof: dA = r dr dθ in polar coordinates: A small polar cell between radii $r$ and $r + \Delta r$ and angles $\theta$ and $\theta + \Delta\theta$ is nearly a rectangle: its radial side has length $\Delta r$ , and its circular side is an arc of radius $r$ subtending $\Delta\theta$ , of length $r …; Double Integrals › Proofs & Why It Matters
Proof: dV = ρ² sin φ dρ dφ dθ: A small spherical cell has three nearly-perpendicular edges: radial, of length $\Delta\rho$ ; along a meridian, an arc of radius $\rho$ subtending $\Delta\phi$ , length $\rho\,\Delta\phi$ ; and along a parallel, an arc of radius $\rho\sin\phi$ (the distance to…; Triple Integrals & Coordinate Systems › Proofs & Why It Matters
Proof: Euler's homogeneous-function theorem: Suppose $f(tx, ty) = t^nf(x, y)$ for all $t$ (homogeneity of degree $n$ ).; The Chain Rule & Implicit Differentiation › Proofs & Why It Matters
Proof: level curves ⊥ gradient: Walk along a level curve $\mathbf r(t)$ with $f(\mathbf r(t)) = c$ constant.; Partial Derivatives & the Gradient › Proofs & Why It Matters
Proof: polar coordinates, this time with no hand-waving: $x = r\cos\theta$ , $y = r\sin\theta$ : $J =; Change of Variables & the Jacobian › Proofs & Why It Matters
Proof: the ½∮(x dy − y dx) area formula: Apply Green's theorem with $P = -\tfrac{y}{2}$ , $Q = \tfrac{x}{2}$ : then $Q_x - P_y = \tfrac12 + \tfrac12 = 1$ , so the loop integral equals $\iint 1\,dA$ — the enclosed area.; Vector Calculus › Proofs & Why It Matters
Proof: the chain rule from the tangent plane: Differentiability means $\Delta z = f_x\Delta x + f_y\Delta y + \varepsilon$ , where $\varepsilon$ shrinks faster than $\sqrt{\Delta x^2 + \Delta y^2}$ .; The Chain Rule & Implicit Differentiation › Proofs & Why It Matters
Proof: the fundamental theorem for line integrals: If $\mathbf F = \nabla f$ and $C$ is parametrized by $\mathbf r(t)$ , $a \le t \le b$ , then $\int_C \mathbf F\cdot d\mathbf r = \int_a^b \nabla f(\mathbf r(t))\cdot\mathbf r'(t)\,dt$ .; Vector Calculus › Proofs & Why It Matters
Proof: the Gaussian integral equals √π: Let $I = \int_{-\infty}^\infty e^{-x^2}dx$ .; Double Integrals › Proofs & Why It Matters
Proof: the gradient is the steepest direction: For a unit vector $\mathbf u$ , the slope in direction $\mathbf u$ is $D_{\mathbf u}f = \nabla f\cdot\mathbf u = |\nabla f|\cos\theta$ , where $\theta$ is the angle between $\mathbf u$ and $\nabla f$ (this dot-product form follows from the chain rule applied …; Partial Derivatives & the Gradient › Proofs & Why It Matters
Proof: the point-to-plane distance formula: Let $Q$ be any point on the plane ( $\mathbf n\cdot Q = d$ ) and $P_0$ the outside point.; Lines & Planes in Space › Proofs & Why It Matters
Proof: the sphere volume formula: Integrate the element over the ball of radius $R$ : $\int_0^{2\pi}\!\!\int_0^{\pi}\!\!\int_0^R \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ .; Triple Integrals & Coordinate Systems › Proofs & Why It Matters
Proof: the tangent plane formula: The plane through $(a, b, f(a,b))$ with slopes $f_x$ in the $x$ -direction and $f_y$ in the $y$ -direction is $z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$ by construction.; Partial Derivatives & the Gradient › Proofs & Why It Matters
Proof: u·v = |u||v|cos θ: Apply the law of cosines to the triangle with sides $\mathbf u$ , $\mathbf v$ , and $\mathbf u - \mathbf v$ : $|\mathbf u - \mathbf v|^2 = |\mathbf u|^2 + |\mathbf v|^2 - 2|\mathbf u||\mathbf v|\cos\theta$ .; Vectors in Space › Proofs & Why It Matters
Proof: why ∇f = λ∇g at a constrained extremum: Let $\mathbf r(t)$ be any curve inside the constraint surface $g = c$ through the extremum at $t = 0$ .; Optimization & Lagrange Multipliers › Proofs & Why It Matters
Proof: why the determinant appears: Near a point, the substitution map is approximated by its derivative — the linear map with the Jacobian matrix $J$ (this is the tangent-plane idea applied to a MAP rather than a function).; Change of Variables & the Jacobian › Proofs & Why It Matters
Proof: κ = |y″|/(1 + y′²)^3/2 for graphs: Curvature is the turning rate of the tangent angle per unit of ARC length: $\kappa = \left|\tfrac{d\phi}{ds}\right|$ where $\tan\phi = y'$ .; Curves & Motion › Proofs & Why It Matters
Rays instead of vertical lines: A region described by $\alpha \le \theta \le \beta$ and $r_1(\theta) \le r \le r_2(\theta)$ has \[_D f dA = ∫_α^β ∫_r_1(θ)^r_2(θ) f(rθ, rθ) r dr dθ.\] The $\theta$ -limits are the directions of th…; Double Integrals › Polar Regions
Reading a contour map: The level curves $f(x, y) = c$ are the contour lines of the surface $z = f(x, y)$ .; Partial Derivatives & the Gradient › Level Curves and the Gradient
Reading spherical limits: In spherical coordinates, $\rho$ is the distance from the origin, $\phi$ the angle down from the positive $z$ -axis ( $0 \le \phi \le \pi$ ) and $\theta$ the longitude.; Triple Integrals & Coordinate Systems › Cones, Shells, and Spherical Limits
Setting up limits: the one skill that decides everything: A double integral over a non-rectangle succeeds or fails at the limits.; Double Integrals › Going Deeper: Explanations & Worked Problems
Solving the Lagrange system, and what λ means: To extremize $f$ subject to $g = c$ , solve $\nabla f = \lambda\nabla g$ together with $g = c$ — for two variables, three equations in $x$ , $y$ , $\lambda$ .; Optimization & Lagrange Multipliers › Lagrange Multipliers in Practice
Split a vector along a direction: The scalar component of $\mathbf u$ along $\mathbf v$ is the signed length $\mathrm{comp}_{\mathbf v}\mathbf u = \dfrac{\mathbf u\cdot\mathbf v}{|\mathbf v|}$ , and the vector projection is $proj_ v u = u· v…; Vectors in Space › Projections, Components, and Work
Substitution in several variables: One-variable substitution carries a factor $\tfrac{dx}{du}$ ; in several variables the factor is a DETERMINANT: for a map $(u, v) \mapsto (x(u,v), y(u,v))$ , $dx\,dy = \left|\det\dfrac{\partial(x, y)}{\partial(u, v)}\right|\,du\,dv$ , where the Jacobian matrix…; Change of Variables & the Jacobian › The Jacobian Determinant
The angle lives in the dot product: Draw both vectors from one point and the angle $\theta$ between them is captured algebraically by $\mathbf u\cdot\mathbf v = |\mathbf u||\mathbf v|\cos\theta$ .; Vectors in Space › Seeing It: Vectors in 2D & 3D
The chain rule through a change of variables: If $z = f(x, y)$ and $x = x(s, t)$ , $y = y(s, t)$ , then \[ $\frac{\partial z}{\partial s}$ = $\frac{\partial z}{\partial x}$ $\frac{\partial x}{\partial s}$ + $\frac{\partial z}{\partial y}$ $\frac{\partial y}{\partial s}$ , $\frac{\partial z}{\partial t}$ = \…; The Chain Rule & Implicit Differentiation › Several Inputs, Several Paths
The chain rule: one term per path: When $z = f(x, y)$ and both $x$ and $y$ depend on $t$ , the total rate of change adds one contribution per route of dependence: $\dfrac{dz}{dt} = \dfrac{\partial z}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial z}{\partial y}\dfrac{dy}{dt}$ .; The Chain Rule & Implicit Differentiation › Differentiating Through Dependencies
The cross product in 3D: perpendicular, with area for length: In three dimensions $\mathbf u\times\mathbf v$ points PERPENDICULAR to the shaded parallelogram that $\mathbf u$ and $\mathbf v$ span — choose which of the two perpendicular directions by the right-hand rule: curl the right hand’s fingers from $\mathbf u$ t…; Vectors in Space › Seeing It: Vectors in 2D & 3D
The cross product measures the patch: If $S$ is parametrized by $\mathbf r(u, v)$ over a region $D$ , then $dS = |\mathbf r_u\times\mathbf r_v|\,du\,dv$ and \[_S F· d S = _D F( r(u, v))·( r_u× r_v) du dv,\] with $ r_u…; Surface Integrals, Stokes & Divergence › Surface Integrals Through a Parametrization
The divergence theorem and Stokes’ theorem: DIVERGENCE THEOREM: outward flux through a closed surface equals $\iiint \nabla\cdot\mathbf F\,dV$ over the enclosed solid — total outflow equals the sum of all the little sources inside.; Surface Integrals, Stokes & Divergence › The Capstone Theorems
The gradient points uphill: The vector $\nabla f = (f_x, f_y)$ points in the direction of STEEPEST increase, and its length is that steepest slope.; Partial Derivatives & the Gradient › Partial Derivatives
The scalar triple product is a volume: For three vectors, $ u·( v× w) =; Vectors in Space › Triple Products and Volume
The tangent plane as a calculator: Near $(a, b)$ , \[f(a + Δ x, b + Δ y) ≈ f(a, b) + f_x(a, b) Δ x + f_y(a, b) Δ y.\] The differential $df = f_x\,dx + f_y\,dy$ is this change, computed on the tangent plane instead of on the surface.; Partial Derivatives & the Gradient › Linear Approximation and Differentials
The volume factor is a 3 × 3 determinant: For a map $(u, v, w)\mapsto(x, y, z)$ , \[dx dy dz = | $\frac{\partial(x, y, z)}{\partial(u, v, w)}$ |du dv dw,\] the determinant of the $3\times3$ matrix of all nine partial derivatives.; Change of Variables & the Jacobian › Jacobians in Three Dimensions
Three nests, three coordinate systems: Boxes: integrate $dz\,dy\,dx$ straight through.; Triple Integrals & Coordinate Systems › Integrating in Three Dimensions
Vector addition is a parallelogram: Placing $\mathbf v$ ’s tail at $\mathbf u$ ’s head (or vice versa — the parallelogram shows both orders at once) lands at the same corner: $\mathbf u + \mathbf v$ .; Vectors in Space › Seeing It: Vectors in 2D & 3D
Velocity, speed, and why arc length is an integral: A moving point $\mathbf r(t)$ has two different "how fast" questions hiding in it.; Curves & Motion › Going Deeper: Explanations & Worked Problems
What the dot product really is, from scratch: Start with lengths: in 3D, $|\mathbf u| = \sqrt{u_1^2 + u_2^2 + u_3^2}$ — Pythagoras applied twice, first in the floor plane to get the horizontal run $\sqrt{u_1^2+u_2^2}$ , then in the vertical plane with the height $u_3$ .; Vectors in Space › Going Deeper: Explanations & Worked Problems
Why critical points, and what can go wrong: At an interior maximum or minimum, every slice through the point is a one-variable function with an extremum there, so every directional derivative vanishes — hence $\nabla f = \mathbf 0$ .; Optimization & Lagrange Multipliers › Going Deeper: Explanations & Worked Problems