Vector function: In the study of mathematics, the concept of a function has been introduced as the correspondence in the set of ordered pairs (x,y) by the rule that for each element x of a set X, we have a unique element y of a set Y, called a function of x and we write y= f(x)
With this idea , we shall now define a vector function.
If to each element of a set of real number R , there corresponds an element r of a vector space V , then we write r=r(t) and call it as a vector function in one independent real variable. The set R is called domain and the set V is called codomain of the function.
Similarly,
Vector Point function: we can define a vector point function in 3 independent variables , if the domain E is a set of points of ordered triads i.e. a region in space . Thus if to each pont (x,y,z) of a domain E there corresponds a vector r(x,y,z) , then r(x,y,z) or simply r is called a vector point function or vector function of position.
For example: the velocity at any point (x,y,z) within a moving fluid at a certain time is a vector point function.
For each point (x,y,z) on the space curve we can get a vector point function as:
r= xi+yj+zk
Or r= r(t)= x(t)i+y(t)j+z(t)k
Differentiation of a vector function: let r(t) and r(t+∆t) be two vector functions of two neighbouring values of t. Put ∆r= r(t+∆t)-r(t).
Then ∆r/∆t = r(t+∆t)-r(t)/∆t
Or
lim∆t-0 ∆r/∆t = lim∆t-0 r(t+∆t)-r(t)/∆t
If the limit exist , then lim∆t-0 ∆r/∆t = dr(t)/dt = dr/dt is called the derivative of r with respect to t. The derivative of a vector function is a vector.
If c(t) is a constant vector function i.e. a constant vector.
Then c(t+∆t) = c(t) and dc/dt = lim∆t-0 c(t+∆t)-c(t)/∆t =0 , a null vector .
Thus the derivative of a constant vector is a null vector.
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