3. Now suppose that (a1, b), (a2, b2),..., (ae, 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 R2 X (the set R2 without the points in X), with the following properties: (i) curl F; = 0 on R? _X for all i = 1, ..., l. (ii) (linearly independent") If C1, C2, ...,Ce are real numbers such that the vector field cFi + c2F2 + ... +ceF is conservative on R2 \X, then c = C2 = ...= ce = 0. (iii) (a spanning set") If G is a vector field on R X with curl G = 0 at all points on R2X, then there are real numbers aj, a2, ..., aq such that G a F 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 3. Now suppose that (a1, b), (a2, b2),..., (ae, 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 R2 X (the set R2 without the points in X), with the following properties: (i) curl F; = 0 on R? _X for all i = 1, ..., l. (ii) (linearly independent") If C1, C2, ...,Ce are real numbers such that the vector field cFi + c2F2 + ... +ceF is conservative on R2 \X, then c = C2 = ...= ce = 0. (iii) (a spanning set") If G is a vector field on R X with curl G = 0 at all points on R2X, then there are real numbers aj, a2, ..., aq such that G a F 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