, , ..., all of a common variable t, so that: or taken together: then the parametrized n-tuple, describes a one-dimensional space curve. Tangent line to curve At a point for some constant t = c, the equations of the one-dimensional tangent line to the curve at that point are given in terms of the ordinary derivatives of r1(t), r2(t), ..., rn(t), and r with respect to t: Normal plane to curve The equation of the n-dimensional hyperplane normal to the tangent line at r = a is: or in terms of the dot product: where are points in the plane, not on the space curve. Relation to kinematics The physical and geometric interpretation of dr(t)/dt is the "velocity" of a point-like particle moving along the path r(t), treating r as the spatial position vector coordinates parametrized by time t, and is a vector tangent to the space curve for all t in the instantaneous direction of motion. At t = c, the space curve has a tangent vector , and the hyperplane normal to the space curve at t = c is also normal to the tangent at t = c. Any vector in this plane (p − a) must be normal to . Similarly, d2r(t)/dt2 is the "acceleration" of the particle, and is a vector normal to the curve directed along the radius of curvature. Matrix valued functions A matrix can also be a function of a single variable. For example, the rotation matrix in 2d: is a matrix valued function of rotation angle of about the origin. Similarly, in special relativity, the Lorentz transformation matrix for a pure boost (without rotations): is a function of the boost parameter β = v/c, in which v is the relative velocity between the frames of reference (a continuous variable), and c is the speed of light, a constant. Banach and Hilbert spaces and quantum mechanics Generalizing the previous section, the output of a function of a real variable can also lie in a Banach space or a Hilbert space. In these spaces, division and multiplication and limits are all defined, so notions such as derivative and integral still apply. This occurs especially often in quantum mechanics, where one takes the derivative of a ket or an operator. This occurs, for instance, in the general time-dependent Schrödinger equation: where one takes the derivative of a wave function, which can be an