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Tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. In other words, a tangent vector at the point x is a linear derivation of the algebra defined by the set of germs at x.
Definition
Let \( f:\mathbb{R}^n\rightarrow\mathbb{R} \) be a differentiable function and let\( \mathbf{v} \) be a vector in \( \mathbb{R}^n \) . We define the directional derivative in the\( \mathbf{v} \) direction at a point \mathbf{x}\in\mathbb{R}^n \) by
\( D_\mathbf{v}f(\mathbf{x})=\left.\frac{d}{dt}f(\mathbf{x}+t\mathbf{v})\right|_{t=0}=\sum_{i=1}^{n}v_i\frac{\partial f}{\partial x_i}(\mathbf{x})\,. \)
The tangent vector at the point \mathbf{x} may then be defined as
\( \mathbf{v}(f(\mathbf{x}))\equiv D_\mathbf{v}(f(\mathbf{x}))\,. \)
Properties
Let \( f,g:\mathbb{R}^n\rightarrow\mathbb{R} \) be differentiable functions, let\( \mathbf{v},\mathbf{w} \) be tangent vectors in \( \mathbb{R}^n \) at \(\mathbf{x}\in\mathbb{R}^n, \) and let \( a,b\in\mathbb{R} \) . Then
\( (a\mathbf{v}+b\mathbf{w})(f)=a\mathbf{v}(f)+b\mathbf{w}(f)
\mathbf{v}(af+bg)=a\mathbf{v}(f)+b\mathbf{v}(g)
\mathbf{v}(fg)=f(\mathbf{x})\mathbf{v}(g)+g(\mathbf{x})\mathbf{v}(f)\,. \)
References
Gray, Alfred (1993), Modern Differential Geometry of Curves and Surfaces, Boca Raton: CRC Press.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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