- #1
kasse
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- 1
To show that a vector field F=(P,Q,R) is conservative, is it enough to show that DP/DY = DQ/DX?
matt grime said:What is the definition of conservative (and just so we're clear: I know what it is, I'm not asking for my benefit)?
matt grime said:Clearly not. R^3 has x,y, and z. You can't just ignore the z. Where did the R go to? I suggest you operate with a nicer definition of conservative, for R^3, such as its curl is 0.
I mean only showing that a certain vector field in R3 is conservative.
A conservative vector field is a type of vector field in which the line integral of the vector field over any closed path is equal to zero. This means that the work done by the vector field on any particle moving along a closed path is equal to zero, regardless of the path taken.
To determine if a vector field is conservative, you can use the fundamental theorem of calculus. If the vector field can be written as the gradient of a scalar function, then it is conservative. Alternatively, you can also calculate the line integral of the vector field over a closed path and see if it equals zero.
Conservative vector fields have many applications in physics, engineering, and mathematics. They are used in the study of fluid dynamics, electromagnetism, and gravity. In engineering, they are used in the design of machines and structures, as well as in optimization problems.
The significance of conservative vector fields lies in the fact that they represent a type of force field that conserves energy. This means that the work done by the force is path-independent and the total energy of the system is conserved. This makes them useful in many real-world applications and also simplifies mathematical calculations.
Yes, a vector field can be partially conservative. This means that the vector field is conservative in some regions, but not in others. In such cases, the vector field can be decomposed into a conservative part and a non-conservative part. The conservative part can be calculated using the methods mentioned in the answer to question 2.