Mathematical analysis is the systematic study of real- and complex-valued functions. A large part of mathematics can be considered as mathematical analysis, including differentiation, integration, dynamical system, series and functional analysis.
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Real function $f: \mathbb{R} \mapsto \mathbb{R}$ is a mapping between subsets of the real line. Real-valued function $f: X \mapsto \mathbb{R}$ is a mapping whose codomain is the real line. Function of a real variable $f: \mathbb{R} \mapsto X$ is a mapping whose domain is the real line.
Zero set $f^{-1}(0)$ of a function $f: X \mapsto \mathbb{R}^k$ is the preimage of zero. Sublevel set $f^{-1}(-\infty, c]$ of a real-valued function on a set $X$ is the preimage of an upper-bounded interval $(-\infty, c]$.
Partial derivative $\frac{\partial f}{\partial x^j}(a)$ of a real-valued function on an open subset of a Euclidean space at a point is its ordinary derivative w.r.t. a coordinate while the other coordinates are fixed: $\frac{\partial f}{\partial x^j}(a) = \lim_{h \to 0} [f(a + h e_j) - f(a)] / h$. Jacobian matrix $(\partial F^i / \partial x^j)$ of a map between (open subsets of) Euclidean spaces is the matrix of the partial derivatives of its component functions.
Continuously differentiable function $f \in C^1(U, \mathbb{R}^m)$ between (open subsets of) Euclidean spaces is one such that all the partial derivatives of its component functions exist and are continuous at each point in its domain. Smooth function $f \in C^\infty(U, \mathbb{R}^m)$ between (open subsets of) Euclidean spaces is one such that all the partial derivatives of all orders of its component functions exist and are continuous at each point in its domain: $\forall i \in \{i\}_{i=1}^m$, $\forall J \in n^k$, $\forall k \in \mathbb{N}$, $\partial^k F^i / \partial x^J \in C^0(U)$. Smooth function between subsets of Euclidean spaces is one that admits a smooth extension in a neighborhood of each point in its domain; a global extension is unnecessary.
Total derivative $DF(a)$ of a map $F: U \mapsto W$ between (open subsets of) finite-dimensional normed vector spaces that is differentiable at an interior point $a$ of its domain is the linear map $DF(a) \in \mathcal{L}(V, W)$ that satisfies $F(a+v) = F(a) + DF(a) v + R(v)$, $\lim_{v \to 0} |R(v)|/|v| = 0$. The total derivative of a function between (open subsets of) Euclidean spaces that is differentiable at an interior point of its domain is the linear map whose standard matrix representation is the Jacobian of the map at that point: $DF(a) = (\partial F^i / \partial x^j)(a)$. Directional derivative $D_v f(a)$ of a smooth real-valued function on an open subset of a Euclidean space at a point $a$ in the direction $v$ is the ordinary derivative of the composite function $g(t) = f(a + t v)$ at zero: $D_v f(a) = d f(a + t v) / d t |_{t=0}$; by chain rule, we have $D_v f(a) = Df(a) v$.
Vector calculus:
Curvilinear coordinates: