Learn Year 2
Multivariable calculus
142 ideas in teaching order. Each idea comes after the ideas that it builds on. 138 of them have a page so far; the others are on the list in grey.
- Euclidean space R^n object
- Function of several variables concept
- Vector-valued function concept
- graph of a function of two variables concept
- Level set definition
- Contour line concept
- Limit of a function of several variables definition
- Limit along different paths concept
- Continuity in R^n definition
- Separately continuous but not continuous function counterexample
- Partial derivative definition
- Directional derivative definition
- Having a directional derivative definition
- Partial derivatives do not imply differentiability counterexample
- Fréchet derivative definition
- Total derivative concept
- Differentiable function of several variables definition
- Differentiable on an open set definition
- Derivative as a little-o approximation statement
- Uniqueness of the derivative statement
- Derivative gives the directional derivatives statement
- Continuous partial derivatives imply differentiability statement
- Derivative of a continuous linear map statement
- Linearity of the derivative statement
- Product rule in several variables statement
- Derivative of a map into a product statement
- Derivative of a map into R^n by components statement
- Chain rule in several variables statement
- Chain rule for a composition with a curve statement
- Gradient definition
- Having a gradient definition
- Gradient points in the direction of steepest ascent statement
- Gradient is orthogonal to level sets statement
- Tangent plane to a graph definition
- Linear approximation concept
- Jacobian matrix definition
- Jacobian determinant definition
- Chain rule for Jacobian matrices statement
- Mean value inequality statement
- Mean value theorem in several variables statement
- Bounded derivative on a convex set implies Lipschitz statement
- Zero derivative on a connected open set implies constant statement
- Higher partial derivatives definition
- Iterated derivative definition
- C^k function characterised by continuous derivatives statement
- Hessian matrix definition
- Symmetry of second derivatives statement
- Schwarz's theorem for C^2 functions statement
- Function with unequal mixed partial derivatives counterexample
- Taylor's theorem in several variables statement
- Second-order Taylor expansion statement
- Multi-index notation concept
- Local maximum and minimum definition
- Derivative vanishes at a local extremum statement
- Saddle point definition
- Second derivative test in several variables statement
- Definite and indefinite quadratic forms concept
- Convex function on R^n definition
- Constrained optimisation concept
- Lagrange multipliers statement
- Lagrange multiplier with one constraint statement
- Strict differentiability definition
- C^1 functions are strictly differentiable statement
- Inverse function theorem statement
- Local homeomorphism from the inverse function theorem statement
- derivative of the inverse function statement
- Implicit function theorem statement
- Implicit function theorem for a surjective derivative statement
- Implicit differentiation concept
- Diffeomorphism definition
- Parametrised curve definition
- Tangent vector of a curve definition
- Regular curve definition
- Arc length parametrisation definition
- Rectifiable curve definition
- Double integral definition
- Triple integral definition
- Iterated integral definition
- Fubini's theorem statement
- Tonelli's theorem statement
- Changing the order of integration statement
- Function for which the iterated integrals differ counterexample
- Integration over a non-rectangular region concept
- Jordan measure definition
- Volume of a box statement
- Cavalieri's principle statement
- Change of variables in multiple integrals statement
- Polar coordinates definition
- Integration in polar coordinates statement
- Cylindrical coordinates definition
- Spherical coordinates definition
- Jacobian of spherical coordinates statement
- Volume of a ball in R^n statement
- Volume of a ball in R^3 statement
- centre of mass definition
- Moment of inertia definition
- Improper multiple integral concept
- Differentiation under the integral sign statement
- Vector field definition
- Line integral of a scalar field definition
- Line integral of a vector field definition
- Line integral as an integral over the parameter interval statement
- Reversing a path changes the sign of a line integral statement
- Additivity of line integrals under concatenation statement
- Work done by a force field concept
- Conservative vector field definition
- Scalar potential definition
- Gradient theorem statement
- Path independence of line integrals statement
- Closed curve definition
- Divergence definition
- Curl definition
- Del operator concept
- Laplacian definition
- Laplacian as a sum of second partial derivatives statement
- Harmonic function definition
- Vector calculus identities statement
- Curl of a gradient is zero statement
- Divergence of a curl is zero statement
- Curl-free vector field on a convex set is conservative statement
- Simply connected domain definition
- Homotopy invariance of line integrals of closed forms statement
- Curl-free field that is not conservative counterexample
- Green's theorem statement
- Divergence theorem on a rectangle in the plane statement
- Area of a region as a line integral statement
- Parametrised surface definition
- Tangent plane and normal vector of a surface definition
- Surface area definition
- Surface integral definition
- Orientable surface definition
- Flux through a surface definition
- Stokes' theorem statement
- Divergence theorem statement
- Differential form definition
- One-form definition
- Wedge product of forms definition
- Exterior derivative definition
- The exterior derivative squares to zero statement
- Closed and exact forms definition
- Poincaré lemma for one-forms on convex sets statement
- Generalised Stokes theorem statement