Here are some definite problems on the exam. Let f: DR be a real valued function defined on an open domain DC Rn. (a) Complete the definition: Given p E D and a vector VER", to say that is a gradient vector for f at p means: (b) Prove the theorem: If V is a gradient vector for f at p then V მე Əx1 მე (p) Jxn I remember some confusion on this topic from earlier in the semester. Unlike in Calc 3, the equation on the line above is not the definition of the gradient vector. Instead, the definition is what you have to finish in (a). (c) Complete the definition: To say that f is differentiable at p ЄD means (d) Prove the theorem: If ƒ is differentiable at p then f is continuous at p. [You can find this proof in my handout on the Chain Rule, found on Canvas]. For these problems, I will also accept proofs for the special case of R² (the proof for R mostly differs dint of having messy notation). Here are some possible problems. You will probably be able to choose some subset of these to write, so you do not necessarily have to prepare all of them for the mid-term. I may be adding to this list by sometime tomorrow (Friday, October 25). (a) Complete the definition: To say that f is a C¹ function on D means (b) Prove the theorem: If ƒ is a C¹ function on D then f is differentiable at every p (a,b) = D. (c) Prove the theorem: Let f: R2 R is a C¹ function, and suppose that the mixed second derivatives and exist on D. Prove Dxdy მყმე 82 that if and მყმ are continuous at p then (p) = Know the theory of conservative vector fields. Jyox (p). Know how to do problems on finding potential functions of vector fields.