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How bad is this statement regarding the Fundamental Theorem for Line Integrals?

  1. Apr 17, 2012 #1
    State the Fundamental Theorem:

    Let F be a vector field.

    If there exists a function f such that F = grad f, then

    [itex]\int_{C} F \cdot dr = f(Q) - f(P)[/itex]

    where P and Q are endpoints of curve C.


    I didn't receive any credit for this answer. Admittedly, it's not very good. I failed to mention that C is a continuous, oriented curve among other things.

    But when I asked my professor about why I received no credit, she opened the textbook and said "THAT is the theorem." She wanted it word for word, claiming that was the difference between asking for the Theorem and definition.

    So, my question is, what is the actual guideline for writing a theorem? Obviously, she wanted it word for word, but as far as I know my textbook doesn't give it work for word what some other textbook would.
  2. jcsd
  3. Apr 17, 2012 #2
    Missing any requirement of the theorem's precept may make the resulting equation invalid. For example, as your statement stands, it is false. Consider F(x) = {(2x)sin(1/x^2) - (2/x)cos(1/x^2), when x is non-zero and 0 when x = 0} and the curve C parametrized by r(t) = x(t) = t, -1 <= t <= 1. F is the gradient of a function, so it fits your theorem, but the equation does not hold, as F is not even integrable on C. Someone who applied your theorem to this case without checking the continuity of F over C would end up with a number that does not mean anything.
    I doubt your teacher expects you to regurgitate the exact words used by a particular author to state a theorem; she probably just wanted you to look at the theorem again and see the concepts that are missing from your statement.
    Last edited: Apr 17, 2012
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