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Jhenrique
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What is a vector valued in a function? For example: ##f(\vec{r})##; or a vector valued in another vector, as: ##\vec{f}(\vec{r})##. What this means? How this kind of calculus is done?
A function is something that maps one space into another. If that other space is a vector space, the function is a "vector-valued function". For example, position as a function of time. The input to the function is time, a scalar. The output is a vector. Thus ##\vec x(t)## is a vector valued function. The input can also be a vector. For example, ##\vec F(\vec r) = -G m_1 m_2 \vec r/||\vec r||^3## also is a vector valued function. On the other hand, a function that maps from a vector space to a scalar space (e.g., potential energy as a function of position) is not a vector valued function.Jhenrique said:What is a vector valued in a function?
With partial derivatives.How this kind of calculus is done?
A vector valued in a function is a mathematical concept where the output of a function is a vector instead of a scalar. This means that the function is able to map a single input to multiple outputs in the form of a vector.
A regular function has a single output for every input, while a vector valued function has multiple outputs for a single input. This means that the output of a vector valued function is a vector, while the output of a regular function is a scalar.
Some examples of vector valued functions include vector fields, parametric equations, and vector-valued differential equations. These types of functions are commonly used in physics, engineering, and computer graphics.
The notation for vector valued functions is similar to regular functions, except the output is represented as a vector. For example, a vector valued function f(x) would be written as f(x) = (f1(x), f2(x), f3(x),...fn(x)).
Vector valued functions have many applications in fields such as physics, engineering, and computer graphics. They are used to model physical phenomena, describe motion of objects, and create 3D graphics. They are also used in calculus to solve problems involving multiple variables.