Download Ordinary derivative If a is regarded as a vector function of a single

Ordinary derivative
If a is regarded as a vector function of a single scalar variable, such as time t, then the equation
above reduces to the first ordinary time derivative of a with respect to t,[1]
[edit] Total derivative
If the vector a is a function of a number n of scalar variables qr (r = 1,...,n), and each qr is only a
function of time t, then the ordinary derivative of a with respect to t can be expressed, in a form
known as the total derivative, as[1]
Some authors prefer to use capital D to indicate the total derivative operator, as in D/Dt. The
total derivative differs from the partial time derivative in that the total derivative accounts for
changes in a due to the time variance of the variables qr.
[edit] Reference frames
Whereas for scalar-valued functions there is only a single possible reference frame, to take the
derivative of a vector-valued function requires the choice of a reference frame (at least when a
fixed Cartesian coordinate system is not implied as such). Once a reference frame has been
chosen, the derivative of a vector-valued function can be computed using techniques similar to
those for computing derivatives of scalar-valued functions. A different choice of reference frame
will, in general, produce a different derivative function. The derivative functions in different
reference frames have a specific kinematical relationship.
[edit] Derivative of a vector function with nonfixed bases
The above formulas for the derivative of a vector function rely on the assumption that the basis
vectors e1,e2,e3 are constant, that is, fixed in the reference frame in which the derivative of a is
being taken, and therefore the e1,e2,e3 each has a derivative of identically zero. This often holds
true for problems dealing with vector fields in a fixed coordinate system, or for simple problems
in physics. However, many complex problems involve the derivative of a vector function in
multiple moving reference frames, which means that the basis vectors will not necessarily be
constant. In such a case where the basis vectors e1,e2,e3 are fixed in reference frame E, but not in
reference frame N, the more general formula for the ordinary time derivative of a vector in
reference frame N is[1]
where the superscript N to the left of the derivative operator indicates the reference frame in
which the derivative is taken. As shown previously, the first term on the right hand side is equal
to the derivative of a in the reference frame where e1,e2,e3 are constant, reference frame E. It also
can be shown that the second term on the right hand side is equal to the relative angular velocity
of the two reference frames cross multiplied with the vector a itself.[1] Thus, after substitution,
the formula relating the derivative of a vector function in two reference frames is[1]
where NωE is the angular velocity of the reference frame E relative to the reference frame N.
One common example where this formula is used is to find the velocity of a space-borne object,
such as a rocket, in the inertial reference frame using measurements of the rocket's velocity
relative to the ground. The velocity NvR in inertial reference frame N of a rocket R located at
position rR can be found using the formula
where NωE is the angular velocity of the Earth relative to the inertial frame N. Since velocity is
the derivative of position, NvR and EvR are the derivatives of rR in reference frames N and E,
respectively. By substitution,
where EvR is the velocity vector of the rocket as measured from a reference frame E that is fixed
to the Earth.
[edit] Derivative and vector multiplication
The derivative of the products of vector functions behaves similarly to the derivative of the
products of scalar functions.[2] Specifically, in the case of scalar multiplication of a vector, if p is
a scalar variable function of q,[1]
In the case of dot multiplication, for two vectors a and b that are both functions of q,[1]
Similarly, the derivative of the cross product of two vector functions is[1]
[edit] Derivative of an n-dimensional vector function
A function f of a real number t with values in the space Rn can be written as
. Its derivative equals
.
If f is a function of several variables, say of
, then the partial derivatives of the
components of f form a
matrix called the Jacobian matrix of f.
[edit] Infinite-dimensional vector functions
If the values of a function f lie in an infinite-dimensional vector space X, such as a Hilbert space,
then f may be called an infinite-dimensional vector function.
[edit] Functions with values in a Hilbert space
If the argument of f is a real number and X is a a Hilbert space, then the derivative of f at a point t
can be defined as in the finite-dimensional case:
Most results of the finite-dimensional case also hold in the infinite-dimensional case too, mutatis
mutandis. Differentiation can also be defined to functions of several variables (e.g.,
or
even
, where Y is an infinite-dimensional vector space).
N.B. If X is a Hilbert space, then one can easily show that any derivative (and any other limit)
can be computed componentwise: if
(i.e.,
space X), and f'(t) exists, then
, where
is an orthonormal basis of the
.
However, the existence of a componentwise derivative does not guarantee the existence of a
derivative, as componentwise convergence in a Hilbert space does not guarantee convergence
with respect to the actual topology of the Hilbert space.
[edit] Other infinite-dimensional vector spaces
Most of the above hold for other topological vector spaces X too. However, not as many classical
results hold in the Banach space setting, e.g., an absolutely continuous function with values in a
suitable Banach space need not have a derivative anywhere. Moreover, in most Banach spaces
setting there are no orthonormal bases.