Rhseung
논문2026.03 – 2026.08

[예시] 다변수 미적분·선형대수 공식 모음

Hyunseung Ryu · GIST

논문저장소

This is a rendering test entry, not a real paper. It strings together standard identities from multivariable calculus and linear algebra (Apostol, 1969; Strang, 2016) purely to exercise equation, matrix, and citation rendering.

Differential calculus in several variables

For f:Rn→Rf : \mathbb{R}^{n} \to \mathbb{R}, the gradient and directional derivative along a unit vector u\mathbf{u} are

∇f(x)=(∂f∂x1,…,∂f∂xn),Duf(x)=∇f(x)⋅u.\begin{align}\nabla f(\mathbf{x})&= \left( \frac{\partial f}{\partial x_{1}}, \ldots, \frac{\partial f}{\partial x_{n}}\right), \\ D_{\mathbf{u}}f(\mathbf{x})&= \nabla f(\mathbf{x}) \cdot \mathbf{u}.\end{align}

For f:Rn→Rm\mathbf{f}: \mathbb{R}^{n} \to \mathbb{R}^{m}, the Jacobian is the matrix of first partials,

Jf(x)=[∂f1∂x1⋯∂f1∂xn⋮⋱⋮∂fm∂x1⋯∂fm∂xn],\begin{equation}J_{\mathbf{f}}(\mathbf{x}) = \begin{bmatrix}\dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n}\end{bmatrix},\end{equation}

and the multivariable chain rule for g(t)=f(x(t))g(t) = f(\mathbf{x}(t)) reads

dgdt=∑i=1n∂f∂xidxidt=∇f(x(t))⋅x′(t).\begin{equation}\frac{dg}{dt}= \sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}\frac{dx_{i}}{dt}= \nabla f(\mathbf{x}(t)) \cdot \mathbf{x}'(t).\end{equation}

The Hessian collects second partials,

Hf(x)=[∂2f∂x12⋯∂2f∂x1∂xn⋮⋱⋮∂2f∂xn∂x1⋯∂2f∂xn2],\begin{equation}H_{f}(\mathbf{x}) = \begin{bmatrix}\dfrac{\partial^2 f}{\partial x_1^2} & \cdots & \dfrac{\partial^2 f}{\partial x_1 \partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial^2 f}{\partial x_n \partial x_1} & \cdots & \dfrac{\partial^2 f}{\partial x_n^2}\end{bmatrix},\end{equation}

and the second-order Taylor expansion about a\mathbf{a} is

f(x)≈f(a)+∇f(a) ⁣⊤(x−a)+12(x−a) ⁣⊤Hf(a)(x−a).\begin{equation}f(\mathbf{x}) \approx f(\mathbf{a}) + \nabla f(\mathbf{a})^{\!\top}(\mathbf{x}- \mathbf{a}) + \frac{1}{2}(\mathbf{x}- \mathbf{a})^{\!\top}H_{f}(\mathbf{a}) (\mathbf{x}- \mathbf{a}).\end{equation}

Vector calculus

For a vector field F=(P,Q,R)\mathbf{F}= (P, Q, R), divergence and curl are

∇⋅F=∂P∂x+∂Q∂y+∂R∂z,∇×F=∣ijk∂x∂y∂zPQR∣.\begin{align}\nabla \cdot \mathbf{F}&= \frac{\partial P}{\partial x}+ \frac{\partial Q}{\partial y}+ \frac{\partial R}{\partial z}, \\ \nabla \times \mathbf{F}&= \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ \partial_{x}&\partial_{y}&\partial_{z} \\ P&Q&R\end{vmatrix}.\end{align}

The three classical integral theorems relate a boundary integral to an integral over the enclosed region:

∮∂D(P dx+Q dy)=∬D(∂Q∂x−∂P∂y)dA(Green),∬S(∇×F)⋅dS=∮∂SF⋅dr(Stokes),∭V(∇⋅F) dV=∯∂VF⋅dS(Divergence).\begin{align}\oint_{\partial D}(P \, dx + Q \, dy)&= \iint_{D} \left( \frac{\partial Q}{\partial x}- \frac{\partial P}{\partial y}\right) dA&&\text{(Green)}, \\ \iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S}&= \oint_{\partial S}\mathbf{F}\cdot d\mathbf{r}&&\text{(Stokes)}, \\ \iiint_{V} (\nabla \cdot \mathbf{F}) \, dV&= \oiint_{\partial V}\mathbf{F}\cdot d\mathbf{S}&&\text{(Divergence)}.\end{align}

Linear algebra

For A∈Rn×nA \in \mathbb{R}^{n \times n}, eigenpairs (λ,v)(\lambda, \mathbf{v}) satisfy

Av=λv,det⁡(A−λI)=0.\begin{equation}A \mathbf{v}= \lambda \mathbf{v}, \qquad \det(A - \lambda I) = 0.\end{equation}

If AA is diagonalizable, A=PDP−1A = P D P^{-1} with D=diag⁡(λ1,…,λn)D = \operatorname{diag}(\lambda_{1}, \ldots, \lambda_{n}), and every real matrix admits a singular value decomposition

A=UΣV ⁣⊤,U ⁣⊤U=I,V ⁣⊤V=I,Σ=diag⁡(σ1,…,σr,0,…).\begin{equation}A = U \Sigma V^{\!\top}, \qquad U^{\!\top}U = I, \quad V^{\!\top}V = I, \qquad \Sigma = \operatorname{diag}(\sigma_{1}, \ldots, \sigma_{r}, 0, \ldots).\end{equation}

A symmetric matrix AA is positive definite iff its quadratic form is strictly positive off the origin,

x ⁣⊤Ax>0∀ x≠0  ⟺  λi>0 for all i.\begin{equation}\mathbf{x}^{\!\top}A \mathbf{x}> 0 \quad \forall \, \mathbf{x}\neq \mathbf{0}\iff \lambda_{i} > 0 \ \text{for all }i.\end{equation}

For an inner product space, the Cauchy–Schwarz inequality bounds the inner product by the norms,

∣⟨x,y⟩∣≤∥x∥ ∥y∥,∥x∥=⟨x,x⟩,\begin{equation}|\langle \mathbf{x}, \mathbf{y}\rangle| \le \lVert \mathbf{x}\rVert \, \lVert \mathbf{y}\rVert, \qquad \lVert \mathbf{x}\rVert = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle},\end{equation}

and the orthogonal projection of y\mathbf{y} onto x\mathbf{x} is

proj⁡xy=⟨x,y⟩⟨x,x⟩ x.\begin{equation}\operatorname{proj}_{\mathbf{x}}\mathbf{y}= \frac{\langle \mathbf{x}, \mathbf{y} \rangle}{\langle \mathbf{x}, \mathbf{x} \rangle}\, \mathbf{x}.\end{equation}

Constrained optimization

For f,g:Rn→Rf, g : \mathbb{R}^{n} \to \mathbb{R}, a critical point of ff subject to g(x)=0g(\mathbf{x}) = 0 satisfies, for some multiplier λ\lambda,

∇f(x∗)=λ ∇g(x∗),g(x∗)=0,\begin{equation}\nabla f(\mathbf{x}^{*}) = \lambda \, \nabla g(\mathbf{x}^{*}), \qquad g(\mathbf{x}^{*}) = 0,\end{equation}

equivalently ∇L=0\nabla \mathcal{L}= \mathbf{0} for the Lagrangian

L(x,λ)=f(x)−λ g(x).\begin{equation}\mathcal{L}(\mathbf{x}, \lambda) = f(\mathbf{x}) - \lambda \, g(\mathbf{x}).\end{equation}

Conclusion

Every identity above is standard (Apostol, 1969; Strang, 2016); the entry exists to check that display equations, aligned systems, matrices, and citations all render correctly, not to report a result.

참고문헌

Apostol, T. M. (1969). Calculus, Volume II: Multi-Variable Calculus and Linear Algebra with Applications (2nd ed.). Wiley.
Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press.