All curves through point p have a tangent vector, not only world lines.

The four-velocity is a tangent vector to the world line.

Let be the basis, tangent vectors in this new coordinates system.

As indicated previously, the 1-jet of a curve through p is a tangent vector.

Rather, it must be chosen so that the tangent vector has constant magnitude.

Such tangent vectors are said to be parallel transports of each other.

We then have that so is a timelike tangent vector with respect to the too.

The classical velocity of an object is a tangent vector to its path.

These two bits of data, a direction and a magnitude, thus determine a tangent vector at the base point.

Another way to think about tangent vectors is as directional derivatives.

The tension parameter, t, changes the length of the tangent vector.

The three tangent vectors at form a basis.

Conditions on the tangent vectors of the curves then define the causal relationships.

Thus the length of a tangent vector cannot be defined, but the angle between two vectors still can.

Let be a closed curve with nowhere-vanishing tangent vector .

Generally speaking, the tangent vector may be zero.

Comparison of tangent vectors at different points on a manifold is generally not a well-defined process.

Though, for a curve whose tangent vectors all have the same sign of magnitude, the arc length is defined.

This angle is equal to the angle between the tangent vectors to the curves.

So all that needs to be proved here is that the map above does indeed define a tangent vector field.

An applet demonstrating parallel transport of tangent vectors on a sphere.

Smooth regular curves (or paths) in can be classified depending on their tangent vectors.

They can be thought of as alternating, multilinear maps on k tangent vectors.

The normal vector is parallel to where is the tangent vector.

From this perspective, let X be a basis of tangent vector fields (not necessarily induced by a coordinate system).