Overview Triple integrals, surface integrals and contour integrals, and bridges between them.

Stokes’ for work (in space) The work done by a vector field along a closed curve, can be replaced with a double integral.

Curl (in space) The curl measures the value of the vector field to be conservative. For a velocity field, curl measures the rotation component of the motion.

Gradient Field (in space) Test for gradient field. Find the potential of a gradient field.

Diffusion/heat (in space) The Diffusion equation governs motion of e.g. smoke in unmovable air, or dye in a solution.

Divergence (in space) Divergence measures how much the flow is expanding. It singles out the stretching component of motion.

Work (in space) Whenever a force is applied to an object, causing the object to move, work is done by the force.

Vector Fields; Surface integrals; Flux (in space) About vectors in space and determining the surface vector, using flux as an example.

Triple Integrals Using triple integrals we can find volume between two surfaces.

Flux; Green’s (in plane) Flux is the amount of something (water, wind, electric field, magnetic field) passing through a surface.

Curl; Green’s (in plane) For a velocity field, curl measures the

*rotation*component of the motion. Curl also measures how far the vector field is from being conservative. Matrices Matrices can be used to express linear relations between variables. For example when we change coordinate systems.

Vectors Vectors do not have a start point, but do have a magnitude (length) and direction. They are described in terms of the unit vectors, or using angle brackets notation.

Gradient Field (in plane) When a vector field is a gradient of function f(x,y), it is called a gradient field.

Vector fields; Line Integrals; Work (in plane) Line integrals in scalar field; line integrals in vector field;.

Double Integrals Find the volume between a function f(x,y) and a certain region in the xy-plane.

Foundations Function graphs; parametric curves.

Overview Vector calculus is about differentiation and integration of vector fields. This article gives an overview of the differentiation. operations.

Quadratic equations Derives the equation for the roots of a general quadratic equation.

Complex Functions [latex][/latex]All our arithmetic functions gracefully extend from the one-dimensional number line into the 2-dimensional \(\mathbb{C}\)-plane. We will introduce the functions that operate on complex arguments.

Complex Numbers Instead of projecting the future merits of complex numbers, we will introduce them in an intuitive way. We draw a parallel to negative numbers that have been universally accepted around the same time.

Linear differential equations Introduces Lineair Inmomogeneous Differential Equations with constant coefficients.

Laurent Series Named after Pierre Alphonse Laurent, a French mathematician and Military Officer, published in the series 1843. The Laurent series is a representation of a complex function f(z) as a series. Unlike the Taylor series which expresses \(f(z)\) as a series of terms with non-negative powers of \(z\), a Laurent series includes terms with negative powers. …

Partial Fraction Expansion Oliver Heaviside (1850-1925), was an English electrical engineer, mathematician and physicist who among many things adapted complex numbers to the study of electrical circuits. He introduced a method to decompose rational function of polynomials as they occur when using the Laplace transform to solve differential equations. Whenever the denominator of a rational function can be …

Binomial Theorem Proof of Isaac Newton generalized binomial theorem

Geometric series Derives formulas for finite and infinite geometric power series.

Complex Arithmetic Formulas Complex arithmetic formulas written in LaTex used in the HP-41 program. Includes everything from power to trigonometric functions.

Euler’s formula Proofs Euler's formula using the MacLaurin series for sine and cosine. Introduces Euler's identify and Cartesian and Polar coordinates.

Complex Arithmetic in Extended Memory for HP-41cv/cx Introduces complex number operations to the HP-41CV and HP-41CX pocket calculators. Easy to use. Magnitude of functions. Runs in extended memory.

Complex Arithmetic with Adjustable Branch Cut for HP-41cv/cx This program introduces complex number operations to the HP-41 calculator. You can adjust the branch cut in the complex plane.