Jacobian Animation, Parametrizing Surfaces, Divergence and Curl. Using Geogebra digital tool.
Math 251, Spring 2024 Geogebra 3 Geogebra is a free digital tool for mathematical learning. If you go to their website https://www.geogebra.org and click on “App Downloads” on the left column, you will notice that you can either download the app, or just click on “Start” if you want to use a particular feature of Geogebra online. The solutions for this assignment should consist of a document (for example a word or pdf file) where you include all the images you generated, plus any steps in the solution of the problem that were used for arriving at these images. Geogebra also allows you to visualize the Jacobian in useful ways. For example, if you enter “Jacobian Animation” on the Geogebra.com search bar, you will find the following animation https://www.geogebra.org/m/HpH5NX7U Exercise 1 Enter the transformation x = u − v, y = u + v. What does the rectangle 0 ≤ u ≤ 2, −1 ≤ v ≤ 1 become on the xy plane? Use the button “animation” and show the uv picture, as well as the xy picture. What is the area of the figure on the xy plane? What is the Jacobian factor? Can you see why the Jacobian is needed in order to correct for the distortion introduced when switching to the uv plane? Parametrizing Surfaces Exercise 2 Parametrize the monkey saddle z = x3 − 3xy 2 by using u = x, v = y as the parameters. On the 3D calculator option of Geogebra use the command “Surface(Expression, Expression, Expression, Parameter Variable 1, Start Value, End Value, Parameter Variable 2, Start Value, End Value)”. Take the values for u, v between −1 and 1. Save the image you get as a pdf or jpeg file. Divergence and Curl For a vector field F(x, y) = hF1 (x, y), F2 (x, y), 0i = F1 (x, y)i + F2 (x, y)j in the plane, the divergence is given by ∂F1 ∂F2 ∇·F= + ∂x ∂y while the curl is given by ∇×F= 0, 0, ∂F2 ∂F1 − ∂x ∂y Since the first two entries in this case will always be zero, we can just focus on the third entry, which is sometimes denoted curlz F. Exercise 3 Find ∇ · F and curlz F for: (a) F = hx, y, 0i (b) F = h−y, x, 0i (c) F = 2xy, y 2 (d) Use the animation on https://www.geogebra.org/m/XfmAAUTG to show the pictures you get for the three previous vector fields. Notice that curlz can be interpreted as the angular velocity of a paddle wheel you place at (x, y), if you think of the vector field as giving you the velocity of a fluid. Save the images you get as pdf or jpeg files.
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