A vector field is a mathematical function that assigns a vector to each point in a given space. It is used to describe the behavior of physical phenomena, such as flow of fluids or electric and magnetic fields.
A path, or curve, in vector calculus is represented by a function c(t) that maps a parameter t to a point in space. This function c(t) is also called a parametric curve.
In vector calculus, a path and a curve refer to the same concept. However, a path is typically used to describe a specific function c(t), while a curve can refer to the general concept of a continuous line in space.
A path is parameterized by choosing a function c(t) that maps a parameter t to a point in space. This parameterization allows us to describe the path in terms of the variable t, making it easier to perform calculations and analyze the behavior of the path.
To find a path c(t) that represents a given curve, you can use the process of integration. First, you need to find the vector field that best describes the behavior of the curve. Then, you can integrate this vector field to obtain the function c(t) that represents the path of the curve.