Example 1.
Compute the pushforward of the vector X1, defined at the point [x = 1, y = 3] by the transformation Phi. Check this answer against the component calculation of the pushforward using the Jacobian matrix.
The entries of the vector C agree with the components of the vector Y1.
Example 2.
Show that the points p1 = [1, 1] and p2 = [- 1, - 1], in M, map under the transformation Phi to the same point in N but that the pushforward of the vector field D_x by Phi at these two points are different.
Example 3.
Compute the pushforward of the vector field X3 by the transformation Phi3 at an arbitrary point. Here we use the second calling sequence for Pushforward.
Example 4.
Express the vector field X4 in polar coordinates. First set up the polar coordinate system and define the transformation Phi4 from polar to Cartesian coordinates. Calculate the inverse transformation. Use the third calling sequence for Pushforward.
Example 5.
Find the tangent vector to the curve t -> [x = t^2, y = t^3, z = t^3].
Example 6.
Find a basis for the tangent plane to the surface z = x^3 - 3*x*y^2 at each point [x, y, z].
Example 7.
Find the projection of the vector field X5 under the map Phi5.
The goal now is to rewrite the coefficients of this vector field in terms of the variables u and v. The map Phi5 is not invertible but it does admit a right inverse Sigma.
We use this map as the second argument in the third calling sequence for Pushforward.
Example 8.
The Pushforward command can also be applied to a list of vectors.