# Multivariable conservative field

• MHB
• kenporock
In summary, a conservative field can be found by considering the force of gravity and moving an object around so that it ends up at the same height it started at. This results in the work done by the force being zero, indicating a conservative field.
kenporock
Goodnight,

How can I find a conservative field F, such that ∫F ds = 0 without C being a closed path
Can i have some examples ?

kenporock said:
Goodnight,

How can I find a conservative field F, such that ∫F ds = 0 without C being a closed path
Can i have some examples ?

Hi kenporock, welcome to MHB! (Wave)

Consider the force of gravity.
Now move an object around, such that it ends up on the same height it had at the beginning.
The work done by gravity $∫\mathbf F\cdot d\mathbf s$ is then zero.
That is, the object has the same potential energy again.

## What is a multivariable conservative field?

A multivariable conservative field is a mathematical concept used in vector calculus to describe a vector field that satisfies certain conditions. In simpler terms, it is a vector field whose line integral is independent of the path taken.

## What are the conditions for a vector field to be considered multivariable conservative?

In order for a vector field to be considered multivariable conservative, it must have a continuous partial derivative with respect to each variable and the curl of the field must be equal to zero.

## What is the significance of a multivariable conservative field in science?

Multivariable conservative fields have many applications in science and engineering, particularly in the fields of fluid mechanics, electromagnetism, and thermodynamics. They allow for the simplification of complex calculations and help to describe physical phenomena in a more precise and efficient manner.

## How is a multivariable conservative field represented mathematically?

A multivariable conservative field is represented using the gradient vector operator (∇) and a scalar function, known as the potential function. The potential function is used to calculate the line integral of the field, which is independent of the path taken.

## What are some real-world examples of multivariable conservative fields?

Examples of multivariable conservative fields in the real world include electric and magnetic fields, fluid flow fields, and gravitational fields. These fields can be described using the principles of multivariable conservative fields and have many practical applications in technology and engineering.

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