Our first manga depicts the conversion of cartesian to polar coordinates. We used an example where the region R inside of a circle of radius 2 is equal to {(x,y) | x^2+ y^2 = 4} and showed how to solve for this region through converting to polar coordinates(Panel 3). We move on to examine the parameters of the circle. Starting with the x parameters, the circle spans from -2 ≤ x ≤ 2 as shown in the circle graph. For the y parameters, we must solve them. To clarify, if we were to say that the y parameters were -2 ≤ y ≤ 2, the area of this region would not look like a circle but a square, which is not what we want. As a result, we can take the equation of the circle and solve for y (Panel 4). However, solving for the area would produce a double integral that is very complicated to solve(Panel 5). This is where we convert our cartesian coordinates (x,y) to polar coordinates (r, θ). In Panel 7, there is a representation of how polar coordinates are produced from cartesian coordinates. The next panels demonstrate how to produce a rectangle to create a simplified integral and concludes comic 1.
The second manga focuses on the concept and real-world application of gradient. We picture a scenario where two people are trying to find the shortest route to reach a certain altitude. We humanize the coordinate (x,y,z) to introduce the concept of gradient in order to “tell” the two people the way they should be heading toward is where the gradient vector points to. The contour map models the mountain using the function m(x), and it visualizes and identifies the group’s location in xy-coordination, and with every contour line marking the points that share the same altitudes (Panel 5). In order to calculate and find the exact direction of gradient and understand which way to go, we have to bring in the concept of partial derivatives of m(x,y) (Panel 8). As pointed out in Panel 11, in the real world, the direction of the steepest ascent may be the most time-saving route, but also causes risks in activities such as mountain climbing. After the “lecture” from X, Y, and Z, the two tourists thanked them and waved goodbye before going on their way, concluding the comic.
