conversations.from: human
conversations.value: a description for a book titled Advanced Calculus Demystified about Ph.D, Pitzer, College with the genre: Science & Math, Mathematics
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conversations.value: <p align="left"><strong>the author</strong>, Ph.D., is an assistant professor of mathematics at Pitzer College in California. He currently teaches courses in multivariable calculus as well as mathematical logic games.
conversations.from: human
conversations.value: Delete the first sentence.
conversations.from: assistant
conversations.value: D., is an assistant professor of mathematics at Pitzer College in California. He currently teaches courses in multivariable calculus as well as mathematical logic games.
conversations.from: human
conversations.value: the first five paragraphs for this book
conversations.from: assistant
conversations.value: Advanced Calculus Demystified
  Notice that the result is just the difference in elevation between your beginning and ending point, and doesn't matter what path you take to get from one to the other!
  The next theorem that relates derivatives, integrals, and vector fields is special to 2. Recall that the curl of a vector field
  To understand what is going on here, we will have to go all the way back to the definition of a multiple integral. Recall the steps to define the value of
  We now move on to Step 4. In this step, we are instructed to compute the sum of the values in Step 3. Using our approximation of each term, we can approximate
  So, the function  is a measure of the "circulation" of **W** around each point. There is a way to experience such a function for yourself. Many people enjoy going "tubing"—floating down a river in an inner tube. Suppose you are tubing and decide you want to stop somewhere to enjoy the scenery, so you drop an anchor. Then you notice that the water on your left is rushing past you faster than the water on your right. What happens? Your inner tube starts to turn. This turning is a measure of the strength of.
conversations.from: human
conversations.value: three more paragraphs
conversations.from: assistant
conversations.value: We know that one way to "differentiate" a vector field is to take its curl. It is then not surprising that the integral of the curl of a vector field should be special. Another way to "differentiate" a vector field in 3 is to take its divergence. In this section, we explore what happens when we integrate the divergence of a vector field. To this end, suppose
  Recall that Green's Theorem gave us a geometric interpretation of  and Stokes' Theorem gave us a geometric interpretation for ∇ × **W**. We now use Gauss' Theorem to give a geometric interpretation of ∇. **W**. This will be completely analogous.
  The magnitude of 〈2, 1〉 is  To get a unit vector that points in the same direction we just divide by the magnitude:
conversations.from: human
conversations.value: Reverse the sentences.
conversations.from: assistant
conversations.value: The magnitude of 〈2, 1〉 is  To get a unit vector that points in the same direction we just divide by the magnitude:. This will be completely analogous. **W**. We now use Gauss' Theorem to give a geometric interpretation of ∇. To this end, suppose
  Recall that Green's Theorem gave us a geometric interpretation of  and Stokes' Theorem gave us a geometric interpretation for ∇ × **W**. In this section, we explore what happens when we integrate the divergence of a vector field. Another way to "differentiate" a vector field in 3 is to take its divergence. It is then not surprising that the integral of the curl of a vector field should be special.We know that one way to "differentiate" a vector field is to take its curl
conversations.from: human
conversations.value: the last four paragraphs
conversations.from: assistant
conversations.value: At the point ψ0(0) = (1, 0, 0) the derivative  is the vector 〈0, 1, 0〉. This agrees with the arrow in the picture.
  At the point ψ0(0) = (1, 0, 1) the derivative  is again the vector 〈0, 1 , 0〉. This is opposite to the arrow in the picture, so we will have to remember to negate the integral over this curve.
  We may thus obtain the desired answer by evaluating the integral on the left-hand side of this last equation. The first thing we will need to evaluate this is the divergence of **W:**
  But the statement of the problem specifies ∇ × **W** = 〈0, 0, 0〉, so the integral on the right (and hence the integral on the left) is 0. Now notice