Contact Info

Location Engineering Building | room 332
Phone 435-797-1494
Contact Christian Bolander
Email christian.bolander@usu.edu

Hours

Mon–Thu	10am – 5pm
Fri	10am – 4pm

Drop-Ins Welcome!

Closed weekends and university holidays.

Reading and Writing Mathematical Expressions

Now mathematics is both a body of truth and a special language, a language more carefully defined and more highly abstracted than our ordinary medium of thought and expression.
- H.B. Williams.

Just as learning a new language requires you to understand symbols, syntax, and context, mathematics is a language the requires effort to learn and decipher. The intent of this module is to help you get comfortable with the language of mathematics by focusing on common symbols and structure that will help you to read mathematical equations as a sentence. Specifically, we also want you to understand how being able to interact with mathematics in this way will be useful to you as an engineer. The following tables show math symbols that are commonly seen in engineering.

Arithmetic Operators

Description	Notation	Example	Reads
Addition	$+$	$a + b$	"a plus b" "the sum of a and b"
Subtraction	$-$	$a - b$	"a minus b" "a subtract b" "the difference of a and b"
Plus/Minus	$\pm$ or $\mp$	$a \pm b$	"a plus or minus b"
Multiplication	$\times$ or $\cdot$	$a \times b$	"a times b" "the product of a and b"
Division	$/$	$a / b$ or $\frac{a}{b}$	"a divided by b" "the quotient of a and b"
Inverse (Multiplicative)	$☐^{-1}$	$a^{-1} = \frac{1}{a}$	"the inverse of a" "a inverse"
Unary Minus	$-☐$	$-a$	"negative a"
Square Root	$\sqrt$	$\sqrt a$	"the square root of a"
Cube Root	$∛$	$∛ a$	"the cubed root of a"
nth Root	ⁿ √	ⁿ √ a	"the nth root of a"
Absolute Value	$\|☐\|$	$\|a\|$	"the absolute value of a"
Exponential	^	$x^{a}$	"x to the a" "x to the power of a"
Percentage	$%$	$10%$	"10 percent"

Functions

Description	Notation	Example	Reads
Function (Morphism)	$\to$ or $\mapsto$	$f: x \to y$	"f is a function from x to y" "f maps x to y"
Image (Range)	$()$ or $[]$	$f(x)$	"f of x"
Free Variable	$(·)$	$f(t, ·)$	"f of t, et cetera"
Inverse (Functional)	$^{-1}$	$f^{-1} : y \to x$	"the inverse of f maps y to x"
Composition	$⚬$	$(g ⚬ f) (x)$	"g of f of x"
Convolution	$*$	$(g * f) (x)$	"the convolution of g and f"
Big O Notation (Order of Magnitude)	$o (☐)$ or $O (☐)$	$f (x) = O (g (x))$	"f of x is of order g of x"

Equality, Equivalence, and Similarity

Description	Notation	Scalar Example	Reads
Equality	$=$	$a = b$	"a equals b"
Inequality	$\neq$	$b \neq c$	"b is not equal to c"
Approximately Equal	$\approx$	$π \approx 3.14$	"pi is approximately equal to 3.14"
Similar To	$\sim$	$a \sim b$	"a is similar to b" "a is of the same magnitude as b"
Identity	$\equiv$	$a + 0 = a$	"a plus zero is equivalent to a"
Proportional To	$\propto$	$cx \propto x$	"cx is proportional to x"
Assignment	$≔$	$a ≔ b$	"a is assigned to b"

Calculus

Description	Notation	Example	Reads
Summation	$\sum_{☐}^{☐}$	$\sum_{i = 1}^{n} a_{i}$	"the sum of a_i from i equals 1 to n"
Product	$\prod_{☐}^{☐}$	$\prod_{i = 1}^{n} z_{i}$	"the product of z_i from i equals 1 to n"
Derivative	$^{'}$ or $^{\cdot}$ or d or ⁽⁾	$\frac{d f}{d x}$	"the derivative of f with respect to x" "d f d x" "d d x of f"
Partial Derivative	$\partial$ or ☐	$\int x$	"the partial of f with respect to x" "partial f partial x"
Antiderivative	$\int$	$\int x d x$	"the integral of x, d x"
Definite Integral	$\int_{☐}^{☐}$	$\int_{a}^{b} x d d$	"the integral of x, d x from a to b"
Line Integral (Curve Integral)	$\oint$	$\oint_{C} f (x, y) d s$	"the line (curve) integral of f d s along C"
Double Integral (Surface Integral)	$\iint$ or $∯$	$\iint_{R} f (x, y) d A$	"the double (surface) integral of f d A over the region R"
Triple Integral (Volume Integral)	$∭$ or $∰$	$∭_{B} f (x, y, z) d V$	"the triple (volume) integral of f d V over the volume B"
Gradient (Nabla)	$\nabla$ or $\nabla^{\to}$ or $del (☐)$ or $grad (☐)$	$\nabla f$	"the gradient of f" "del f" "grad f" "nabla f"
Laplacian	$\nabla^{2}$ or $∆$	$\nabla^{2} f$ or $∆ f$	"the Laplacian of f" "del squared f" "nabla squared f" "Delta f"
Goes To	$\to$	$x \to \infty$	"x goes to $\infty$ "
Infinity	$\infty$	$a \to \infty$	"a goes to infinity"
Limit	$lim_{☐ \to ☐} (☐)$	$lim_{x \to 0} f (x)$	"the limit as x goes to 0 of f of x"
From the Left	$☐^{-}$	$lim_{x \to 2^{-}} f (x)$	"the limit as x goes to 2 from the left of f of x
From the Right	$☐^{+}$	$lim_{x \to a^{+}} g (x)$	"the limit as x goes to a from the right of g of x

See Vector Calculus and Matrix Calculus for more information on how these operations apply to vectors and matrices.

Linear and Multilinear Algebra

Description	Notation	Example	Reads
Vector	$\vec{☐}$ or $\overline{☐}$	$\vec{a}$	"the vector a"
Dot Product	$\cdot$ or $(☐, ☐)$ or $❬☐, ☐❭$	$\vec{u} \cdot \vec{v}$	"u dot v"
Cross Product	$\times$ or $[☐, ☐]$	$\vec{u} \times \vec{v}$	"u cross v"
Triple Product	$(☐, ☐, ☐)$	$(\vec{u}, \vec{v}, \vec{w})$	"the triple product of u, v, and w"
Outer Product (Dyadic Product) (Tensor Product)	$\otimes$	$\vec{a} \otimes \vec{b}$	"the outer (dyadic/tensor) product of a and b"
Magnitude (Euclidean Norm)	$\|☐\|$	$\|\vec{a}\|$	"the magnitude of a"
Norm	$\|\|☐\|\|$	$\|\|\vec{v}\|\|$	"the norm of v"
p-Norm	${\|\|☐\|\|}_{p}$	${\|\|\vec{v}\|\|}_{p}$	"the pth norm of v"
Unit Vector	$\hat{☐}$	$\hat{i}$	"the unit vector i"
Matrix	capital ☐	$A$	"the matrix A"
Kronecker Product	$\otimes$	$A \otimes B$	"the Kronecker product of A and B"
Hadamard (Schur) Product	$⚬$	$A ⚬ B$	"the Haradmard (Schur) product of A and B"
Trace	$t r (☐)$	$t r (A)$	"the trace of A"
Matrix Norm	$\|\|☐\|\|$	$\|\|A\|\|$	"the norm of A"
Determinant	$\|☐\|$	$\|A\|$	"the determinant of A"
Inverse (Multiplicative)	$☐^{-1}$	$A^{-1}$	"the inverse of A" "A inverse"
Pseudoinverse (Moore-Penrose)	$☐^{+}$	$A^{+}$	"the pseudoinverse of A"
Matrix Exponential	^	$e^{A}$	"e to the A"
Transpose	$☐^{⊤}$ or $☐^{T}$	${\vec{v}}^{T}$ or $A^{⊤}$	"v (or A) transpose"
Conjugate Transpose (Hermitian Transpose)	$☐^{*}$ or $☐^{H}$ or $☐^{†}$	${\vec{v}}^{*}$ or $A^{H}$	"v (or A) conjugate transpose"

See Vector Spaces for additional information on the manipulation of vector spaces common in linear algebra.

Statistics and Probability

Description	Notation	Example	Reads
Measure of Probability	$P(☐)$	$P (A)$	"the probability of event A"
Conditional Probability	$P (☐ ∣ ☐)$	$P (A ∣ B)$	"the probability of event A given that B has already occured"
Expected Value	$E (☐)$	$E (X)$	"the expectation of X"
Variance	$V(☐)$	$V(X)$	"the variance of X"
Standard Deviation	$σ (☐)$	$σ (A)$	"the standard deviation of A"
Covariance	$σ (☐, ☐)$	$σ (A, B)$	"the covariance of A and B"
Correlation	$ρ (☐, ☐)$	$ρ (A, B)$	"the correlation between A and B"
Average (Mean)	$\overline{☐}$ or $❬☐❭$	$\overline{x}$	"the average (mean) of x"
Estimator	$\hat{☐}$	$\hat{p}$	"the estimate of p"

Arithmetic Comparison

Description	Notation	Example	Reads
Less Than	<	$a < b$	"a is less than b"
Greater Than	$>$	$c > b$	"c is greater than b"
Less Than or Equal To	$\leq$	$f (x) \leq 5$	"f(x) is less than or equal to 5"
Greater Than or Equal To	$\geq$	$f (x) \geq f (y)$	"f(x) is greater than or equal to f(y)"
Much Less Than (Order of Magnitude)	$≪$	$c ≪ d$	"c is much less than d"
Much Greater Than (Order of Magnitude)	$≫$	$d ≫ a$	"d is much greater than a"

Number Sets

Description	Notation	Example	Reads
Natural Numbers	$ℕ$	$a \in ℕ$	"the set of (all) natural numbers"
Integers	$ℤ$	$b \in ℤ$	"the set of (all) integers"
Rational Numbers	$ℚ$	$d \in ℚ$	"the set of (all) rational numbers"
Real Numbers	$ℝ$	$\vec{v} \in ℝ^{3}$	"the set of (all) real numbers"
Complex Numbers	$ℂ$	$z \in ℂ$	"the set of (all) complex numbers"

Intervals and Mathematical Constants

Description	Notation	Example	Reads
Closed Interval	$[☐, ☐]$	$[a, b]$	"(on) the closed interval from a to b"
Open Interval	$(☐, ☐)$ or ]☐, ☐[	$(a, b)$	"(on) the open interval from a to b"
Pi	$π$	$3 π$	"three pi(e)"
Euler's Number	$e$	$e^{x}$	"e to the x"

Complex Numbers and Combinatorials

Description	Notation	Example	Reads
Imaginary Unit	$i$ or $j$	$a + b j$	"a plus b j"
Complex Conjugate	$\overline{☐}$ or $☐^{*}$	$z^{*}$ or $\overline{z}$	"the complex conjugate of z"
Argument (Polar Coordinates)	$\arg$	$\arg (z)$	"the argument of z"
Modulo	$\mod ☐$	$a \mod b$	"a mod b"
Factorial	$!$	$n!$	"n factorial"
Combination	$(_{☐}^{☐})$	$(_{k}^{n})$	"the combination of n things taken k at a time"

Set Theory

Description	Notation	Example	Reads
In or Belongs To (A Set)	$\in$	$\vec{a} \in ℝ^{3}$	"the vector $\vec{a}$ is in R-3"
Not In or Does Not Belong To (A Set)	$\notin$	$c \notin (a, b)$	"c is not in the interval from a to b"
Empty Set	$\emptyset$	$\vec{x} \notin \emptyset$	"the vector $\vec{x}$ is not in the empty set"
Such That	$∣$ or :	$n ∣ n \in ℤ$	"n such that n is an integer"
Subset	$\subseteq$	$A \subseteq B$	"A is a subset of B" "B contains A" "B includes A"
Proper Subset	$⊊$	$A ⊊ B$	"A is a proper subset of B"
Set Union	$\cup$	$A \cup B$	"the union of A and B"
Set Intersection	$\cap$	$A \cap B$	"the intersection of A and B"
Set Difference	\ or -	$A \ B$ or A - B	"the difference of A and B"
Infimum	$\inf (☐)$	$\inf (A)$	"the infimum of A"
Supremum	$\sup (☐)$	$\sup (B)$	"the supremum of B"

Basic Logic

Description	Notation	Example	Reads
For All (Universial Quantification)	$\forall$	$\forall Ɛ > 0$	"for all epsilon greater than 0"
There Exists ... Such That (Existential Quantification)	$\exists$	$\exists η < 0$	"there exists an eta less than 0"
There Exists Exactly One ... Such That (Uniqueness Quantification)	$\exists!$	$\exists! η < 0$	"there exists exactly one eta less than 0"
There Does Not Exist ... Such That (Existential Quantification)	$∄$	$∄ η < 0$	"there does not exist an eta less than 0"
Implies (Logical Consequence)	$\Rightarrow$	$x / a = 1 \Rightarrow x = a$	"x/a=1 implies that x equals a"
Is Equivalent To If and Only If (Logical Equivalence)	$\Leftrightarrow$ or $iff$	$iff a = b$	"if and only if a equals b"
Not (Logical Negation)	$\neg$ or '	$\neg P$	"not P"
Or (Logical Or)	$⋁$ or +	$R ⋁ S$	"R or S"
And (Logical And)	$⋀$ or ·	$R ⋀ S$	"R and S"
XOR (Exclusive Or)	$\oplus$ or $⊻$	$R \oplus S$	"either R or S, but not both"
Therefore	$∴$	$∴ b = a$	"therefore b equals a"
Because	$∵$	$∵ b < 0$	"because b < 0"
...As Desired (Q.E.D.)	$☐$ or $∎$	$\underset{☐}{c = d}$	"c equals d as desired"

Geometry (Euclidean)

Description	Notation	Example	Reads
Line Segment	$[☐]$ or $\|☐\|$ or $\overline{☐}$ or $\underline{☐}$	$\overline{AB}$	"the line segment AB"
Vector	$\vec{☐}$	$\vec{a}$	"the vector $\vec{a}$ "
Angle	$∠$	$∠ LMN$	"the angle LMN"
Triangle	$∆$	$∆ ABC$	"the triangle ABC"
Parallel	$∥$	$\overline{AB} ∥ \overline{CD}$	"the line segment $\overline{AB}$ is parallel to the line segment $\overline{CD}$ "
Not Parallel	$∦$	$\overline{AB} ∦ \overline{CD}$	"the line segment $\overline{AB}$ is not parallel to the line segment $\overline{CD}$ "
Orthogonal	$⊥$	$\overline{AB} ⊥ \overline{CD}$	"the line segment $\overline{AB}$ is perpendicular to the line segment $\overline{CD}$ "

Engineering Context

MAE: The Central Limit Theorem (Instrumentation)

$$ \sigma_{\overline{x}} = \sigma / \sqrt{n} $$

Steady Flow Mass Balance (Thermodynamics)

$$ \sum_{\mathrm{in}} \dot{m} = \sum_{\mathrm{out}} \dot{m} $$

CEE:

ECE: Boolean Algebra (Digital Circuits)

$$ w = ps'k + t = ((p \land \neg s) \land k) \lor t $$

BE:

A Deeper Dive

Consider the following mathematical statement: Let $f : [a, b] \mapsto ℝ$ be continuous on the closed interval $[a,, b]$ and differentiable on the open interval (a, b) with a < b. Then

$$ \exists \: c \in (a, b) \: | \: f'(c) = \frac{f(b) - f(a)}{b - a} $$

This is a statement for the *mean value theorem* taught in Calculus courses. In engineering, we use this theorem in aerodynamics, fluid mechanics, controls, and in many other areas. We can analyze this statement using the information in the tables above.

The first statement notes that the function f maps from a closed interval from a to b to the set of all real numbers. That means that we can input a number from a to b and will get a real number out of the function. We also learn that the function must be continuous on the interval from a to b and differentiable on the open interval from a to b. We also learn (which we may have already guessed) that a must be greater than b.

So long as our function satisfies these requirements over the intervals we have specified, then the equation itself can be read like this. If these requirements are satisfied, then there exists a number c in the open interval from a to b such that the derivative of the function at c is equal to the function at b minus the function at a divided by b minus a.