Applying vector calculus knowledge griffths and other e& m writers like to use elegant conceptualizations that quickly get you to a result with seemingly very little work. this relies on a deeper understanding of vector calculus, which you will develop in this course. in this problem, you will explain a conceptualization that you come to a result, then you will connect that conceptualization to a mathematical proof. for this, consider the surface integral that we define as the "vector area", da 1. for a hemispherical bowl (a half sphere) of radius r, you will find a. sketch the situation and show the vector da in your sketch. in which direction will a point once the integral is complete? how do you know? 2. what is a reasonable guess for the value of a? why is that a reasonable guess? 3. compute directly the value of a and comment on your guesses from parts 1 and 2. it's ok if your answers to parts 1 and 2 don't match the computed answer. explain how things fit together (e. g., what did you get right? what needs more work on your 4. the value of a takes on the same value for any closed surface. what is a reasonable guess for a in this case? what is your rationale that makes that guess reasonable? 5. show using the gradient theorem what the value of a is for any closed surface. compare your result to your answer to part 4. again, it's ok if your answer to part 4 don't match the computed answer. explain how things fit together (e. g., what did you get right? what needs more work on your conceptualization? )