Basically, every part of mathematics has its own function. Both its mathematical functions, its application in life. No exception with vectors. Mathematically, we sometimes assert that a vector function A (x, y, z) can be understood that a vector field because it associates a vector with every point in a place.

A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head. source

A vector function r(t) = hf(t), g(t), h(t)is a function of one variable—that is, there is only one “input” value. What makes vector functions more complicated than the functions y = f(x) that we studied in the first part of this book is of course that the “output” values are now three-dimensional vectors instead of simple numbers. It is natural to wonder if there is a corresponding notion of the derivative for vector functions. In the simpler case of a function y = s(t), in which t represents time and s(t) is position on a line, we have even that the derivative s ′(t) represents velocity; we might hope that in a similar way the derivative of a vector function would tell us something about the velocity of an object moving in three dimensions. source

Mathematically, it can be explained the function of that vector, If for every scalar value U is associated with a vector A, then A is called a function u denoted by A (u). In three dimensions is written A (u) = A1 (u) i + A2 (u) j + A3 (u)

We sometimes state that a vector function A (x, y, z) defines a vector field because it associates a vector with every point in a place. In the same way, 4 (x, y, z) defines a scalar field because it associates a scalar with each point in an area.

The concept of this function can be easily So we for every point (x, y, z) = A1 (x, y, z) with one vector A, then A is a function (x, y, z) and with A (x, y, z) i + A2 (x, y, z) j + A3 (x, y, z) k.

The vector function r : I → R n, with n = 2, 3, has a limit given by the vector L when t approaches t0, denoted as lim t→t0 r(t) = L, iff: For every number > 0 there exists a number δ > 0. source

Limit, continuity and derivative vector functions follow similar rules for the scalar function in question. The following statement shows the similarities that exist.

1. The vector function A (u) is said to be continuous at u0 if given a positive number, we can determine a positive number. So <when <. This is equivalent to the statement = A (u0).
2. The derivative of A (u) is defined as provided that this limit exists. If A (u) = A1 (u) i + A2 (u) j + A3 (u) k. Then, The same concept will apply to higher derivatives like and so on.

Examples of vector functions, for example, the equations of the free movement of a particle in space. If each point in a space (R3) is associated with a vector, then space is called a vector field. Examples of vector fields, such as fluid flow (gas, heat, water and so on) in a room.

Any function not associated with a vector is called a scalar function, and a space with no vertex associated with a vector is called a scalar field. Examples of scalar fields, such as the temperature of any point in a chamber or iron rod, at a time.

Conclusion

Vector is a quantity that has value and direction. To declare a vector can be done on a plane or Cartesian XOY coordinate field by drawing a line segment with arrows at one end. The length of the line segment represents the length (length) of the vector and the arrow represents the direction of the vector. Vectors are symbolized in bold or with underlined letters.

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Reference

Vector_introductionsource
Vector_Functions source
Teaching source

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