Overview Triple integrals, surface integrals and contour integrals, and bridges between them.
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.
Diffusion/heat (in space) The Diffusion equation governs motion of e.g. smoke in unmovable air, or dye in a solution.
Work (in space) Whenever a force is applied to an object, causing the object to move, work is done by the force.
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.
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.
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. …
Z-transform Proofs Proofs for Z-transform properties, pairs, initial and final value. Includes derivative, binomial scaled, sine and other functions.
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
Inverse Z-transform The inverse Z-transform, can be evaluated using Cauchy's integral. Which is an integral taken over a counter-clockwise closed contour C in the region of converge of (z)
Discrete Transfer Functions Transfer function in the Z-domain let us determine the discrete system response characteristics without having to solve the underlying equations.
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.