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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 R n. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of ...
Tangent vectors are described in the differential geometry of curves in the context of curves in R n. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally, a tangent vector at the point is a linear derivation of the algebra defined by the ...
Illustration of tangential and normal components of a vector to a surface. In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector.
Every continuously differentiable curve in M can be lifted to a curve in F in such a way that the tangent vector field of the lifted curve is the lift of the tangent vector field of the original curve. This statement means that any frame on a curve can be parallelly transported along the curve. This is precisely the idea of "moving frames".
The normal curvature, k n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ r, measures the rate of change of the surface ...
Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach .
The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and its angle is zero. In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
The exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see tangent space). If tangent vectors are defined as equivalence classes of the curves γ {\displaystyle \gamma } for which γ ( 0 ) = x , {\displaystyle \gamma (0)=x,} then the differential is given by