Is the Vector Field F(x,y) Conservative?

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In summary, we are given a vector field F=(e^x sin(y) + tan(y))i + (e^x cos(y) + sec^2(y))j and asked to determine if it is conservative (exact). After applying the necessary derivatives, we find that the vector field is not conservative. However, if the vector field were F=(e^x sin(y) + tan(y))i + (e^x cos(y) + xsec^2(y))j, it would be conservative. Another way to check for exactness is to take the curl of the vector field, which is not possible in this case as it is a purely two-dimensional problem.
  • #1
Dx
Determine whether the vector field is conservative. F(x,y)=(e^x sin(y) + tan(y)i + (e^x cos(y) + sec^2(y)j.
f(x1,y1) = [inte]A to B (F * T) ds = [inte]A to B(e^x sin(y) + tan(y)dx + (e^x cos(y) + sec^2(y)dy = [inte]1 to 0 (e^x sin(y) + tan(y)dx + (e^x cos(y) + sec^2(y)dy

I am lost from here can anyone help me solve from here please?
Dx
 
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  • #2
You are given that F=(e^x sin(y) + tan(y))i + (e^x cos(y) + sec^2(y))j.

The rest of the formula you gave is not necessary. F is the vector field. saying that it is "conservative" (that's really a physics term- the mathematics term is "exact") means that it is the gradient of some scalar function. In other words, is there a function f(x,y) such that F= grad(f)= (df/dx)i+ (df/dy)j.

If df/dx= e^x sin(y)+ tan(y) then

df/dxdy= e^x cos(y)+ sec^2(y)

If df/dy= e^x cos(y)+ sec^2(y)

df/dydx= e^x cos(y). Notice that df/dxdy is not the same as df/dy/dx: but mixed derivatives have to be equal! F is not the gradient of any function, this vector field is not conservative (exact).

By the way, if it were F= (e^x sin(y) + tan(y))i + (e^x cos(y) + xsec^2(y))j then it would be conservative. Do you see the difference?
 
  • #3
Or even more directly, if you take the curl of the vector field (something we physicsts do all the time), you'll find conservative field curls are equal to 0. Of course, the curl is little computationally intensive, but the answer is unmistakeable and there is a great deal of satisfaction when you get through the thing :smile:
 
  • #4
I wouldn't call that "more directly". What taking the curl involves is taking the derivatives as above, then subtracting to see if the result is 0 rather than comparing to see if they are the same.
I would even point out that the curl requires working in three dimensions while this problem is purely two dimensional.

You are right that is the same thing. And anyone who talks in terms of "conservative vector fields" rather than "exact differentials" might be more comfortable with "curl" than second derivatives.
 
  • #5
I stand corrected. Sometimes in physics, we have difficulty seeing the forest from the trees. THanx
 

1. What is a conservative field?

A conservative field is a vector field in which the line integral along any closed path is equal to zero. In other words, the work done by the field on a particle moving along a closed path is independent of the path taken.

2. What are some examples of conservative fields?

Some common examples of conservative fields include gravitational fields, electric fields in electrostatics, and magnetic fields in magnetostatics. These fields are considered conservative because the work done by them on a particle depends only on the starting and ending points of the particle's path.

3. How is a conservative field different from a non-conservative field?

A conservative field is different from a non-conservative field in that the work done by a non-conservative field on a particle depends on the path taken by the particle. In other words, the line integral along a closed path is not equal to zero for a non-conservative field.

4. What is the significance of conservative fields in physics?

Conservative fields are important in physics because they follow the principle of conservation of energy. This means that the work done by a conservative field on a particle can be fully recovered as the particle moves back to its starting point. This principle is essential in many physical phenomena, such as the motion of planets and the behavior of electric charges.

5. How can conservative fields be represented mathematically?

Conservative fields can be mathematically represented by a vector field, in which each point in space is associated with a vector. This vector represents the direction and strength of the field at that point. In addition, conservative fields can also be described using mathematical equations, such as the gradient of a scalar potential function.

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