- #1
Za Kh
- 24
- 0
I know that if a vector field is conserved then there exits a function such that the gradient of this function is equal to the vector field but am just curious to know the reason of it.
Hi! I've figured this out, by the stocks theorem and that the work done by a conservative force on a closed curve is zero but why it isn't true for the non conservative force?BvU said:Hi ZA,
You of course googled conservative vector field but there must be something that isn't clear to you. What specifically ?
A conserved vector field is a type of vector field in which the divergence is equal to zero. This means that the vector field is "conserved" or does not change over time. A gradient, on the other hand, is a mathematical operation that takes a scalar function and produces a vector field. When a vector field is conserved, it can be shown mathematically that it must be the gradient of a certain function, known as the potential function.
The fact that a conserved vector field is a gradient has important implications in physics and engineering. It allows us to describe the behavior of a system using a single scalar function instead of a complex vector field. This makes calculations and analysis much simpler and more efficient.
In physics, forces that are conservative are those that do not dissipate energy. This means that the work done by the force is independent of the path taken. A conserved vector field is closely related to conservative forces because it describes a force field in which the work done is path independent. This is why conservative forces are often referred to as "conserved" forces.
No, a vector field cannot be conserved without being a gradient. This is because the mathematical definition of a conserved vector field requires the divergence to be equal to zero, and it can be shown that this is only possible if the vector field is a gradient.
Conserved vector fields have many practical applications in physics and engineering. They are used to model and analyze various physical systems, such as fluid flow, electromagnetic fields, and gravitational fields. They also play a crucial role in the study of conservative forces and the conservation of energy in physical systems.