Engineering Calculus, made clear.

Statement, geometric meaning and verification problems.

The slope of a chord equals the slope of a tangent somewhere between.

The mean value theorem for two functions at once.

When a list of numbers settles down to a limit.

Tests to decide whether an infinite sum has a finite value.

Writing functions such as eˣ, sin x and log(1 + x) as power series.

Limits of the form 0/0 and ∞/∞ using derivatives.

0 · ∞, ∞ − ∞, 1^∞, 0⁰ and ∞⁰, reduced to a form L'Hôpital's rule can handle.

Differentiating with respect to one variable at a time.

Why the path of approach matters, and when a function is continuous.

Expanding f(x, y) about a point.

Stationary points, saddle points and the rt − s² test.

Maxima and minima subject to a constraint.

The direction in which a scalar field increases fastest.

The rate of change in any chosen direction.

How a vector field spreads out and rotates, and when it has a potential.

Integrals over infinite ranges, or with an integrand that blows up.

Definition, properties and why Γ(n + 1) = n!.

Definition, standard forms and properties.

The relation between them, and evaluating integrals with it.

Integrating over a region in the plane.

Redrawing the region to make an integral easier.

When circles make Cartesian limits painful.

Areas of regions bounded by curves.

Integrating over a solid region.

Triple integrals over spheres and cones.

Triple integrals over cylinders and paraboloids.

Volumes of solids bounded by surfaces.

Work done by a force along a curve.

A line integral round a curve as a double integral inside it.

The flux of a vector field through a surface.

Circulation round a curve equals the flux of the curl.

Flux out of a closed surface equals the integral of the divergence inside.