About This Page
This page introduces basic mathematical notation common to all departments.
For more advanced mathematics (vector analysis, Fourier transforms, etc.), see the Advanced Science/Engineering page.
Table of Contents
Partial Derivatives
Partial Derivative Notation
Code:
$\displaystyle\frac{\partial f}{\partial x}$
Display:
$\displaystyle\frac{\partial f}{\partial x}$
Second-Order Partial Derivative
Code:
$\displaystyle\frac{\partial^2 f}{\partial x^2}$
Display:
$\displaystyle\frac{\partial^2 f}{\partial x^2}$
Mixed Partial Derivative
Code:
$\displaystyle\frac{\partial^2 f}{\partial x \partial y}$
Display:
$\displaystyle\frac{\partial^2 f}{\partial x \partial y}$
Jacobian
Code:
$\displaystyle J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$
Display:
$\displaystyle J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$
Various Types of Integration
Double Integral
Code:
$\displaystyle\iint_D f(x,y) \, dx \, dy$
Display:
$\displaystyle\iint_D f(x,y) \, dx \, dy$
Triple Integral
Code:
$\displaystyle\iiint_V f(x,y,z) \, dx \, dy \, dz$
Display:
$\displaystyle\iiint_V f(x,y,z) \, dx \, dy \, dz$
Line Integral
Code:
$\displaystyle\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \, dt$
Display:
$\displaystyle\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \, dt$
Surface Integral
Code:
$\displaystyle\iint_S \vec{F} \cdot d\vec{S} = \iint_D \vec{F} \cdot \vec{n} \, dS$
Display:
$\displaystyle\iint_S \vec{F} \cdot d\vec{S} = \iint_D \vec{F} \cdot \vec{n} \, dS$
Linear Algebra
Matrix Multiplication
Code:
$AB$
Display:
$AB$
Identity Matrix
Code:
$I$ または $E$
Display:
$I$ または $E$
Eigenvalue
Code:
$\lambda$
Display:
$\lambda$
Characteristic Equation
Code:
$\det(A - \lambda I) = 0$
Display:
$\det(A - \lambda I) = 0$
Trace
Code:
$\text{tr}(A)$
Display:
$\text{tr}(A)$
Rank
Code:
$\text{rank}(A)$
Display:
$\text{rank}(A)$
Probability and Statistics (University Level)
Conditional Probability
Code:
$\displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)}$
Display:
$\displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)}$
Bayes' Theorem
Code:
$\displaystyle P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
Display:
$\displaystyle P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
Covariance
Code:
$\text{Cov}(X, Y) = E[(X - \mu_X)(Y - \mu_Y)]$
Display:
$\text{Cov}(X, Y) = E[(X - \mu_X)(Y - \mu_Y)]$
Correlation Coefficient
Code:
$\displaystyle\rho = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$
Display:
$\displaystyle\rho = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$
Normal Distribution PDF
Code:
$\displaystyle f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
Display:
$\displaystyle f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
Central Limit Theorem
Code:
$\displaystyle \frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} N(0, 1)$
Display:
$\displaystyle \frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} N(0, 1)$
Moment Generating Function
Code:
$M_X(t) = E[e^{tX}]$
Display:
$M_X(t) = E[e^{tX}]$
Logical Symbols
Universal Quantifier
Code:
$\forall x \in A$ (for all x)
Display:
$\forall x \in A$ (for all x)
Existential Quantifier
Code:
$\exists x \in A$ (there exists an x)
Display:
$\exists x \in A$ (there exists an x)
Series Expansions
Taylor Series
Code:
$\displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$
Display:
$\displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$
Maclaurin Series
Code:
$\displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$
Display:
$\displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$
Vector Calculus
Gradient (grad)
Code:
$\displaystyle\nabla f = \mathrm{grad}\, f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$
Display:
$\displaystyle\nabla f = \mathrm{grad}\, f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$
Divergence (div)
Code:
$\displaystyle\nabla \cdot \vec{F} = \mathrm{div}\, \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$
Display:
$\displaystyle\nabla \cdot \vec{F} = \mathrm{div}\, \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$
Curl (rot)
Code:
$\displaystyle\nabla \times \vec{F} = \mathrm{rot}\, \vec{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)$
Display:
$\displaystyle\nabla \times \vec{F} = \mathrm{rot}\, \vec{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)$
Usage Examples in Reading Notes
Example 1: Partial Derivative Problem
Code:
p.56の問題: $f(x,y) = x^2 + 3xy + y^2$のとき、$\displaystyle\frac{\partial f}{\partial x} = 2x + 3y$、$\displaystyle\frac{\partial f}{\partial y} = 3x + 2y$
Display:
p.56の問題: $f(x,y) = x^2 + 3xy + y^2$のとき、$\displaystyle\frac{\partial f}{\partial x} = 2x + 3y$、$\displaystyle\frac{\partial f}{\partial y} = 3x + 2y$
Example 2: Eigenvalue Calculation
Code:
行列$A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$の固有値は$\det(A - \lambda I) = 0$より$\lambda = 1, 3$
Display:
行列$A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$の固有値は$\det(A - \lambda I) = 0$より$\lambda = 1, 3$
Example 3: Statistics Calculation
Code:
データ$X$が正規分布$N(50, 100)$に従うとき、平均$\mu = 50$、標準偏差$\sigma = 10$
Display:
データ$X$が正規分布$N(50, 100)$に従うとき、平均$\mu = 50$、標準偏差$\sigma = 10$
Common Symbol Reference
\partial→ 偏微分記号∂\iint, \iiint→ 多重積分\forall, \exists→ 全称∀・存在∃\det→ 行列式det\text{}→ 数式内にテキスト