Foundations

Locate a point from an ordered pair, read named axes and their scales, and connect the same values in words, tables and plots. Keep signs, fractions and units exact, distinguish an axis point from a quadrant point, and separate selected observations from a stated rule and its domain before drawing connections.

Read a straight line as a stated relationship, calculate its slope from consistent signed coordinate changes, and interpret its intersections with the axes. Keep horizontal, vertical and coincident-point cases distinct, attach units and domain conditions to context models, and use numerical scales rather than appearance alone to judge a line’s change.

Read a graph’s quantities, units, ticks and context before judging its visual impression. Calculate differences, reference-based percentages and interval rates from the represented values; distinguish legitimate zoom from distorted spacing; and keep samples, interpolation, missing information, uncertainty and causal explanations separate.

Describe displacement with fixed-axis components, combine compatible vectors, and use small exact examples to introduce span, independence, basis and real vector spaces.

In a Euclidean right triangle, squared leg lengths sum to the squared hypotenuse length.

Sine and cosine are unit-circle coordinates whose squared values sum to one.

A complex number has real coordinates a and b and an imaginary unit satisfying i²=−1.

A linear map preserves linear combinations and has a matrix once bases are chosen.

Vector-space laws extend beyond coordinate arrows.

A basis spans without linear redundancy.

Kernel and image dimensions sum to domain dimension.

An eigenvector changes only by a scalar factor.

Self-adjoint operators have orthonormal eigenbases.

Any real rectangular matrix has orthogonal factors.

An invertible affine transformation of ℝⁿ has form T(x)=Ax+b with A invertible.

Every distance-preserving bijection of ℝⁿ has form T(x)=Qx+b with QᵀQ=I; conversely each such map is an isometry.

A bijection of ℝⁿ uniformly scaling all distances by s>0 is a similarity and has form T(x)=sQx+b with Q orthogonal.

The symmetries of a figure S⊂ℝⁿ are the Euclidean isometries g with g(S)=S.

For a one-to-one C¹ transformation T mapping a planar or spatial integration region G onto R, with nonzero interior Jacobian, ∫R f(x)dx=∫G f(T(u))|det DT(u)|du.

A regular C¹ plane or space curve r(t), a≤t≤b, has length ∫ₐᵇ‖r′(t)‖dt.

A C¹ regular parameterization r(u,v) has rᵤ×rᵥ≠0 and area element dS=‖rᵤ×rᵥ‖du dv.

Along a piecewise C¹ regular path r(t), scalar integration is ∫C f ds=∫f(r(t))‖r′(t)‖dt and vector integration is ∫C F·dr=∫F(r(t))·r′(t)dt.

If F=∇f with f of class C¹, then ∫C F·dr=f(endpoint)−f(start), independent of the piecewise smooth path.

For planar F=(P,Q) of class C¹ on a neighborhood of a bounded region D, ∮∂D(P dx+Q dy)=∫D(Qₓ−Pᵧ)dA and outward flux ∮∂D(P dy−Q dx)=∫D(Pₓ+Qᵧ)dA.

Specified matching sides and angles can establish the same triangle shape and size.

Triangle similarity keeps corresponding angles and one common positive side scale.

Opposite and adjacent legs depend on the chosen acute angle.

Radian measure is arc length divided by radius.

Area measures covered surface; perimeter measures boundary length.

Volume and surface area measure different aspects of a solid.