Question

3. Now suppose that (a,b), (a2, b2),..., (aq, be) are l distinct points on R2. Let X be the set formed by these l points. Prove that there are l vector fields F1, F2,..., Fe, each defined on R2X (the set R2 without the points in X), with the following properties: (i) curl F; = 0 on RP X for all i = 1, ..., l. (ii) (“linearly independent”) If C1,C2, ..., Ce are real numbers such that the vector field CzFz+c2F2 + ... +ceF2 is conservative on R2X, then cı = C2 = ... =ce = 0. = 0 at all points on (iii) (“a spanning set") If G is a vector field on R2 X with curl G R? _X, then there are real numbers 01, 02, ..., de such that G - a Fı – a2F2 –... – a Fe is conservative on R2X. Instructive note: The above series of statements show that if X is a set of l distinct points on R2, then the dimension of the 1st De Rham cohomology vector space of R2 X is l. Calculations like this are fundamental in the study of differential geometry and algebraic topology.

Answer 1

Let us look at the vector fields

- a; Fi(n,y) = (-7- a.)2 + (v - b;)2(x - a;)2 + (y - 6:2 y-bi Fi(0,y) = (_

i) All of these fields have curl 0 in .

R²X

ii) Note that if we take any smooth closed curve around these l points, then the loop integral of each of these fields would be positive. Therefore, if

$\sum_{i} c_i \mathbf F_i$

is a conservative field, then it's loop integral through that loop should be 0. Which would imply that all the $c_i$'s should be 0

iii) Let the loop integral of G around a smooth closed curve which encloses X be a. Then we can write a as a linear combination of the loop integrals of $\mathbf F_i$'s. Therefore, we can find an linear combination of $\mathbf F_i$ such that subtracting it from G would have loop integral 0 for any smooth closed loop in $\mathbb R^2$. Then it would be a conservative field in $\mathbb R^2$ (i.e. it would have a potential function).