An ice cream cone is described using rectangular coordinates: z = √8 − x2 − y2 and the cone z = √x2 + y2. (a) Description was transformed into polar coordinates using the transformation f (x, y) = (r cos θ, r sin θ) with x2 + y2 = r2 and tan θ = y/x . State that description of the surfaces in polar coordinates once more. (b) Given a transformation T : x = g(u, v), y = h(u, v), the Jacobian determinant or Jacobian of T is: (see image) Find the Jacobian for the transformation from rectangular coordinates to polar coordinates, i.e. find J(r, θ) using the formulation above. (c) To do a general change of variables for a double integral from rectangular coordinates to another coordinates system using the transformation T : x = g(u, v), y = h(u, v) over the same region R, we can call upon the formula: ∫ ∫ f (x, y) dy dx =∫ ∫ f (g(u, v), h(u, v)) · |J(u, v)| du dv R R Use the above change of variables formula and your answers to the previous parts to set up a double integral to find the volume of the ice cream cone in polar coordinates, i.e. using the transformation T : x = r cos θ, y = r sin θ.
An ice cream cone is described using rectangular coordinates: z = √8 − x2 − y2 and the cone z = √x2 + y2.
(a) Description was transformed into polar coordinates using the transformation f (x, y) = (r cos θ, r sin θ) with x2 + y2 = r2 and tan θ = y/x . State that description of the surfaces in polar coordinates once more.
(b) Given a transformation T : x = g(u, v), y = h(u, v), the Jacobian
determinant or Jacobian of T is: (see image)
Find the Jacobian for the transformation from rectangular coordinates to polar
coordinates, i.e. find J(r, θ) using the formulation above.
(c) To do a general change of variables for a double integral from rectangular coordinates to another
T : x = g(u, v), y = h(u, v) over the same region R, we can call upon the formula:
∫ ∫ f (x, y) dy dx =∫ ∫ f (g(u, v), h(u, v)) · |J(u, v)| du dv R R Use the above change of variables formula and your answers to the previous parts to
set up a double integral to find the volume of the ice cream cone in polar coordinates,
i.e. using the transformation T : x = r cos θ, y = r sin θ.
(d) Interpret the meaning of |J(u, v)| in the change of variables process.
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