Engineering Mathematics

Polar Coordinates

A coordinate system representing points in the real cartesian plane using the distance from the origin $r$ and the direction from the origin expressed as an angle $\theta$, written as $(r, \theta)$.

Rectangular Coordinates

Another name for cartesian coordinates $(x, y)$ which describe a position relative to the $x$- and $y$-axes.

Converting Polar to Cartesian Coordinates

The formulas used to find cartesian values from polar coordinates: $x = r \text{cos}(\theta)$ and $y = r \text{sin}(\theta)$.

Converting Cartesian to Polar Coordinates

The formulas used to find polar values from cartesian coordinates: $r=\sqrt{x^2+y^2}$ and $\text{tan}(\theta) = \frac{y}{x}$.

Function arctan2

A function in the Python library numpy that takes $x$ and $y$ as separate arguments and provides the angle $\theta$ in the correct quadrant corresponding to polar coordinates. Function is called by np.arctan2(y,x).

Gradient of the tangent to a polar curve

The formula for the slope of a curve $r = f(\theta)$ in cartesian terms: $\frac{dy}{dx} = \frac{f'(\theta) \text{sin}(\theta) + f(\theta) \text{cos}(\theta)}{f'(\theta) \text{cos}(\theta) - f(\theta) \text{sin}(\theta)}$.

Complex Numbers ($C$)

Numbers of the form $a + bj$ where $j^2 = -1$ (thus $j=\sqrt{-1}$ ) and $a, b \in R$.

Real Part ($Re$)

The component $a$ of a complex number $a + bj$, denoted as $Re(a + bj) = a$.

Imaginary Part ($Im$)

The component $b$ of a complex number $a + bj$, denoted as $Im(a + bj) = b$.

Purely Imaginary Numbers

Complex numbers where the real part $a = 0$, resulting in the form $bj$.

Addition of Complex Numbers

The operation defined as $(a + bj) + (c + dj) = (a + c) + (b + d)j$.

Multiplication of Complex Numbers

The operation defined by $(a + bj)(c + dj) = (ac - bd) + (ad + bc)j$.

Complex Conjugate

For a complex number $z = a + bj$, the conjugate is denoted as $\bar{z} = a - bj$.

Modulus of a complex number ($|z|$)

The distance of the point representing $z = a + bj$ from the origin in the complex plane, calculated as $|z|=\sqrt{\overline{z}z}=\sqrt{a^2+b^2}$ .

Polar Form of a Complex Number

Representing a complex number as $z = r e^{j\theta}$ , where $r=|z|$ is the modulus and $\theta$ is the argument.

Argument ($\text{arg } z$)

The angle $\theta$ in the polar representation of a complex number, which is determined within a range of $2\pi$ .

Principal Argument ($Arg z$)

The unique value of the argument $\theta$ chosen such that -\pi<\theta<\pi

cmath.polar and cmath.rect()

A specialized Python package used to convert complex numbers into polar form or back to cartesian (rectangular) form.

De Moivre’s Theorem

A theorem stating that for any complex number $z = r e^{j\theta}$, the $n^{th}$ power is calculated as $z^n = r^n e^{jn\theta}$, for $n = 1, 2, \text{...}$.

Impedance ($Z$)

The complex representation used in electrical circuits to relate voltage $V$ and current $I$ via a form of Ohm's Law: $V = IZ$.

Resistor Impedance ($Z_R$)

Determined by the resistance $R$, where $Z_R = R$.

Capacitor Impedance ($Z_C$)

Determined by capacitance $C$ and angular frequency \text{\omega}, where $Z_{C}=\frac{1}{jC\omega}=-\frac{j}{C\omega}$ .This represents the opposition to current flow in a capacitor, indicating that the impedance is inversely related to both capacitance and frequency.

Inductor Impedance ($Z_L$)

Determined by inductance $L$ and angular frequency \text{\omega}, where $Z_{L}=jL\omega$ .

Series Total Impedance

The sum of impedances for $n$ components in series: $Z = Z_1 + Z_2 + \text{...} + Z_n$.

Parallel Total Impedance

The impedance for $n$ components in parallel given by $Z = \frac{1}{\frac{1}{Z_1} + \frac{1}{Z_2} + \text{...} \frac{1}{Z_n}}$.

Limit of a Function

The value that a function approaches as the input approaches a given point.

One-Sided Limit

Limits from one side only, denoted limx→a+ for right-hand limits and limx→a− for left-hand limits.

Local Maxima

Points in the domain where the function value is greater than or equal to the values of the function in some neighborhood.

Local Minima

Points in the domain where the function value is less than or equal to the values of the function in some neighborhood.

Critical Point

A point where the derivative is either zero or undefined.

Second Derivative Test

A method to classify critical points by examining the sign of the second derivative.

Stationary Point

A point where the derivative is zero and the function may be at a local maximum or minimum.

Implicit Differentiation

A technique used to differentiate equations that define y implicitly as a function of x.

Related Rates

Problems involving the rates at which two or more related variables change.

Partial Derivatives

Derivatives of functions with multiple variables, holding other variables constant.

Chain Rule for Partial Derivatives

A formula for finding the derivative of a composite function in the context of multiple variables.

Product Rule

If u and v are functions of x, then the derivative of their product is u'v + uv'.

Quotient Rule

If u and v are functions of x, then the derivative of their quotient is (u'v - uv') / v².

Continuous Function

A function without breaks, jumps, or holes in its graph.

Graph of a Function

The visual representation of the set of ordered pairs (x, f(x)) in a Cartesian plane.

Value Approaching Infinity

Describes the behavior of a function as the input grows larger without bound.

Slope of a Tangent Line

The instantaneous rate of change of a function at a given point, equal to the derivative at that point.

Concave Up

Describes a shape of a graph where the second derivative is positive.

Concave Down

Describes a shape of a graph where the second derivative is negative.

Optimization Problem

A problem that seeks to maximize or minimize a particular quantity.

Chain Rule for Functions

Used for differentiating composite functions, stating that the derivative of f(g(x)) is f'(g(x)) * g'(x).

Exponential Growth

Growth that occurs when a quantity increases at a rate proportional to its current value.

Logarithmic Growth

Growth characterized by a slower rate of increase as the quantity becomes larger.

Limit Existence

A limit is said to exist if the function approaches the same value from all directions.

Non-Continuous Function

A function that has discontinuities; it does not have a well-defined limit at certain points.

Inflection Point

A point on a curve where the concavity changes from up to down or down to up.

Step Function

A piecewise constant function that jumps from one value to another at specific points.

Asymptotic Analysis

A method for analyzing the behavior of functions as they approach certain limits.

Maximum Rate of Change

The highest point at which the variable experiences a change with respect to another variable.

Slope Field

A graphical representation of the solutions to a first-order differential equation.

Geometrical Representation of Vectors

Vectors are represented as line segments with an arrowhead indicating direction.

Equal Vectors

Two vectors are equal if they have the same length and direction.

Zero Vector

The vector whose initial and terminal points are the same; has a length of 0.

Dot Product

A scalar obtained from two vectors, calculated as the product of their magnitudes and the cosine of the angle between them.

Cross Product

A vector resulting from multiplying two vectors, which is perpendicular to the plane containing them. Calculated by the product of their magnitudes and the sine of the angle between them.

Unit Vector

A vector with a length of 1.

Gradient Vector

A vector consisting of the partial derivatives of a function, indicating the direction of maximum increase.

Directional Derivative

The rate of change of a function in the direction of a unit vector.

Parametric Curve

A curve defined by parameters where the x and y coordinates are given as functions of a variable.

Torque

The rotational equivalent of a linear force, calculated as the cross product of the position vector and the force vector.

Angle Between Vectors

Calculated using the dot product formula, $\theta={arccos}\left(\frac{u \cdot v}{||u|| ||v||}\right)$ .

Orthogonal Vectors

Vectors that are perpendicular to each other; their dot product is zero.

Impulse

Change in momentum produced by a force acting over time.

Circle Parametrization

The representation of a circle using parametric equations such as $(cos(t), sin(t))$.

Max Directional Derivative

The maximum directional derivative is equal to the magnitude of the gradient; it occurs in the direction of the gradient.

Arc Length

The length of the path traced by a curve over a specific parameter interval.

Sets of Numbers

Categories of numbers that includes natural numbers, integers, rational numbers, and real numbers.

Natural Numbers (N)

The positive whole numbers: {1, 2, 3, 4, …}.

Integers (Z)

All whole numbers including both positive and negative numbers and zero: {…, -3, -2, -1, 0, 1, 2, 3, …}.

Rational Numbers (Q)

Numbers that can be expressed as fractions of two integers: e.g. 1/2, -5/4.

Real Numbers (R)

All rational numbers and other numbers that cannot be expressed as fractions, including √2 and π.

Element of a Set

A number that belongs to a set, denoted by the symbol ∈.

Subset

A set containing some, but not all, elements of another set. Denoted by $\subset$

Interval Notation

A method to represent subsets of real numbers, indicating ranges between two values.

Closed Interval [a, b]

Set of all x such that a ≤ x ≤ b.

Open Interval (a, b)

Set of all x such that a < x < b.

Union of Sets (∪)

Combines two sets to include all elements from both.

Function

A rule that assigns each element in a domain D a unique value in a set S.

Domain of a Function (D)

The set of all possible inputs for a function.

Range of a Function (R(f))

The set of all output values produced by the function.

Rational Function

A function expressed as the ratio of two polynomial functions, $p(x)/q(x)$, where q(x) ≠ 0.

Exponential Function

Functions of the form $f(x) = a^x$ where a is a positive real number.

Logarithmic Function

The inverse of an exponential function, defining the exponent required to achieve a value.

Trigonometric Functions

Functions such as sin, cos, and tan, relating angles to ratios of side lengths in right triangles.

Sinusoidal Wave

A mathematical curve that describes a smooth periodic oscillation.

Amplitude (A)

Half the distance between the maximum and minimum values of a wave.

Period (T)

The length of time it takes for one complete cycle of a wave.

Frequency (f)

The number of cycles that occur in a unit of time, measured in Hertz (Hz).

Phase Shift

The horizontal shift of the periodic wave, affecting its starting point.

Piecewise Function

A function defined by multiple sub-functions, each applying to a specific interval.

Heaviside Function

A piecewise function defined as H(x) = 0 for x < 0, 1 for x ≥ 0.

Absolute Value Function

A function defined as f(x) = |x| = x if x ≥ 0 and f(x) = -x if x < 0.

Hyperbolic Functions

Functions defined in terms of the exponential function, such as sinh, cosh, and tanh.

Combination of Functions

The result of applying operations (addition, subtraction, multiplication, and division) to functions.

Composite Functions (f ◦ g)

Combining two functions where the output of one function becomes the input of another.