Differential
Undergraduate
Consider a differentiable function $f$ and a point $x$ in its domain. The differential of $f$ at $x$ is the linear map $\mathrm{d}f_x : v \mapsto Df(x) v$ that best approximates $f(x + v) - f(x)$ for small $v$. For $f : \reals^n \to \reals$,
\[\mathrm{d}f_x(v) = \frac{\partial f}{\partial x_1}(x) v_1 + \dots + \frac{\partial f}{\partial x_n}(x) v_n.\]
Graduate (differential geometry)
Consider a smooth function $f : M \to \reals$ on a manifold $M$. The differential of $f$ is the one-form $\mathrm{d}f$, which assigns to each point $x$ the covector $\mathrm{d}f_x$ on the tangent space $T_x M$.
Examples
Worked examples
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$f(t, x) = t x^2$
: The linear map $\mathrm{d}f_{(t, x)} : (\Delta t, \Delta x) \mapsto x^2 \Delta t + 2 t x \Delta x$
Undergraduate
The differential of $f(t, x) = t x^2$ combines both partial derivatives into the linear approximation $Df(t, x)$.
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$f(x, y) = x^2 + 3y$
: $\mathrm{d}f_{(x, y)} : (v_1, v_2) \mapsto 2x v_1 + 3 v_2$
Undergraduate
A linear map that takes a direction and returns a rate of change.
Distinctions involving this term
- Types of multivariable derivatives: The directional derivative, the Jacobian, the gradient, the differential, and the Lie derivative all package the partial derivatives of a function differently: as a vector (gradient), a matrix (Jacobian), a linear map (differential), or a single number in a chosen direction (directional derivative) or along a vector field (Lie derivative).