Jacobians and Transformation of Integrals - Question Bank

1. If a transformation is given by x = u, y = v, what is the Jacobian ∂(x, y) / ∂(u, v)?
A) 1
B) 0
C) -1
D) uv
2. What is the Jacobian determinant for the transformation from Cartesian coordinates (x, y) to polar coordinates (r, θ), where x = r cos(θ) and y = r sin(θ)?
A) r
B) -r
C) r^2
D) 1/r
3. The Jacobian of a transformation from (u, v) to (x, y) is given by the determinant of the matrix:
A) [[∂x/∂u, ∂x/∂v], [∂y/∂u, ∂y/∂v]]
B) [[∂u/∂x, ∂u/∂y], [∂v/∂x, ∂v/∂y]]
C) [[∂x/∂u, ∂y/∂u], [∂x/∂v, ∂y/∂v]]
D) [[∂u/∂x, ∂y/∂x], [∂u/∂v, ∂y/∂v]]
4. Consider the transformation x = u, y = v/u. If we integrate f(x, y) = x over the region 1 ≤ x ≤ 2, 1 ≤ y ≤ x, we would transform this to the uv-plane. The Jacobian is 1. The transformed region is:
A) 1 ≤ u ≤ 2, 1 ≤ v ≤ u
B) 1 ≤ u ≤ 2, 1/u ≤ v ≤ 1
C) 1 ≤ u ≤ 2, 1 ≤ v ≤ u^2
D) 1 ≤ u ≤ 2, 1 ≤ v ≤ u/2
5. If x = u, y = v, and z = w, the Jacobian for the transformation from (u, v, w) to (x, y, z) is:
A) 1
B) 0
C) -1
D) uvw
6. The purpose of the absolute value of the Jacobian in the transformation formula for integrals is:
A) To ensure the scaling factor is positive, as area/volume cannot be negative.
B) To account for the sign change in the determinant.
C) To simplify the integrand.
D) To match the units of the integrand.
7. What is the Jacobian of the transformation x = u cos(v), y = u sin(v)?
A) u
B) u^2
C) cos(v)
D) sin(v)
8. Consider the transformation x = u^2 - v^2, y = 2uv. What is ∂(x, y) / ∂(u, v)?
A) 4(u^2 + v^2)
B) 2(u^2 + v^2)
C) 4uv
D) 2uv
9. The Jacobian determinant is found by calculating the determinant of the matrix whose entries are the partial derivatives of the new coordinates with respect to the old coordinates.
A) False (it's old with respect to new)
B) True
C) False (it's second partial derivatives)
D) False (it's the sum of partial derivatives)
10. If a region R in the xy-plane is defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and we use the transformation x = u, y = uv, the Jacobian ∂(x, y) / ∂(u, v) is u. The transformed region S in the uv-plane is bounded by:
A) u=0, u=1, v=0, v=1
B) u=0, u=1, v=0, v=infinity
C) u=0, u=1, v=0, v=u
D) u=0, u=1, v=0, v=1/u
11. What is the Jacobian of the transformation x = e^u, y = e^v?
A) e^(u+v)
B) e^u * e^v
C) e^u + e^v
D) e^(u-v)
12. When transforming a triple integral using Jacobians, the volume element dx dy dz is replaced by:
A) |∂(x, y, z) / ∂(u, v, w)| du dv dw
B) ∂(x, y, z) / ∂(u, v, w) du dv dw
C) du dv dw
D) |∂(u, v, w) / ∂(x, y, z)| du dv dw
13. The Jacobian of the transformation x = u, y = v sin(u) is:
A) v cos(u)
B) v sin(u)
C) cos(u)
D) sin(u)
14. Consider the transformation x = u - v, y = u + v. What is ∂(x, y) / ∂(u, v)?
A) 2
B) -2
C) 0
D) 1
15. If a transformation maps a small area dA in the (u, v) plane to an area dA' in the (x, y) plane, then dA' is related to dA by:
A) dA' = |∂(x, y) / ∂(u, v)| dA
B) dA' = |∂(u, v) / ∂(x, y)| dA
C) dA' = dA
D) dA' = dA / |∂(x, y) / ∂(u, v)|
16. What is the Jacobian of the inverse transformation from spherical coordinates (ρ, θ, φ) to Cartesian coordinates (x, y, z)?
A) 1/(ρ^2 sin(φ))
B) ρ^2 sin(φ)
C) 1/ρ
D) ρ
17. The transformation of an integral from Cartesian to polar coordinates uses the Jacobian:
A) r
B) 1/r
C) r^2
D) 1
18. Consider the transformation x = u, y = v^2. What is ∂(x, y) / ∂(u, v)?
A) 2v
B) v^2
C) u
D) 1
19. What is the Jacobian of the transformation x = u, y = v, z = w?
A) 1
B) 0
C) -1
D) uvw
20. For a linear transformation T(u, v) = (au + bv, cu + dv), what is the Jacobian ∂(x, y) / ∂(u, v)?
A) ad - bc
B) a + d
C) a - d
D) bc - ad
21. Which of the following transformations would likely be simplified by using Jacobians?
A) Integrating x^2 + y^2 over an elliptical region.
B) Integrating x over a rectangular region.
C) Integrating y over a triangular region.
D) Integrating x^2 over a square region.
22. If x = r cos(θ) and y = r sin(θ), what is the integral of the function f(x, y) = x^2 + y^2 over a disk of radius R centered at the origin, using polar coordinates?
A) ∫ from 0 to 2π ∫ from 0 to R (r^2) * r dr dθ
B) ∫ from 0 to 2π ∫ from 0 to R r^2 dr dθ
C) ∫ from 0 to 2π ∫ from 0 to R r dr dθ
D) ∫ from 0 to R ∫ from 0 to 2π r^2 dr dθ
23. The determinant of the Jacobian matrix represents the local scaling factor for area or volume under a coordinate transformation.
A) True
B) False
C) Only for area scaling
D) Only for volume scaling
24. What is the Jacobian of the transformation x = u, y = v, z = w/uv?
A) -1/(uv)
B) 1/(uv)
C) w/(uv)
D) 1
25. Consider the transformation x = u, y = u + v. What is ∂(x, y) / ∂(u, v)?
A) 1
B) u
C) v
D) u+v
26. The transformation ∫∫_R f(x, y) dx dy can be simplified if the region R is complex, by choosing a transformation T such that the corresponding region S in the (u, v) plane is simpler. This is a primary motivation for using Jacobians.
A) True
B) False
C) This only applies to line integrals.
D) This only applies to triple integrals.
27. If the Jacobian of a transformation is zero, it implies that the transformation is:
A) Singular or non-invertible in the local region.
B) Linear.
C) Conformal.
D) A rotation.
28. What is the Jacobian of the transformation x = u + v, y = w, z = u - v?
A) -2
B) 2
C) 0
D) 1
29. The Jacobian determinant for the transformation to spherical coordinates (ρ, θ, φ) where x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), z = ρ cos(φ) is:
A) ρ^2 sin(φ)
B) ρ^2 cos(φ)
C) ρ sin(φ)
D) ρ cos(φ)
30. Consider the transformation x = u, y = v/u. What is ∂(x, y) / ∂(u, v)?
A) 1/u
B) 1
C) u
D) v
31. When using Jacobians for integration, the region of integration in the new coordinate system is determined by:
A) Transforming the original region's boundaries.
B) The Jacobian determinant itself.
C) The integrand's properties.
D) The limits of the original integral.
32. What is the Jacobian of the transformation from cylindrical coordinates (r, θ, z) to Cartesian coordinates (x, y, z)?
A) r
B) r^2
C) 1
D) z
33. If a transformation is defined by x = u^2, y = v^2, what is the Jacobian ∂(x, y) / ∂(u, v)?
A) 2uv
B) 4uv
C) 2u
D) 2v
34. The Jacobian is crucial for ensuring that the 'area element' or 'volume element' is correctly scaled during a change of variables in multiple integrals.
A) True
B) False
C) Only for 3D integrals
D) Only for 2D integrals
35. Consider the transformation x = 2u, y = 3v. What is the Jacobian ∂(x, y) / ∂(u, v)?
A) 6
B) 5
C) u
D) v
36. When transforming an integral over a region R to an integral over a region S using a transformation T, the formula is:
A) ∫∫_R f(x, y) dx dy = ∫∫_S f(x(u, v), y(u, v)) |J(u, v)| du dv
B) ∫∫_R f(x, y) dx dy = ∫∫_S f(x(u, v), y(u, v)) / |J(u, v)| du dv
C) ∫∫_R f(x, y) dx dy = ∫∫_S f(u, v) |J(u, v)| du dv
D) ∫∫_R f(x, y) dx dy = ∫∫_S f(x, y) du dv
37. What is the Jacobian of the inverse transformation, ∂(u, v) / ∂(x, y), if ∂(x, y) / ∂(u, v) = J?
A) 1/J
B) -1/J
C) J
D) 0
38. If x = u, y = v, z = w, what is the Jacobian of the transformation?
A) 1
B) 0
C) -1
D) uvw
39. The transformation of a triple integral ∫∫∫_V f(x, y, z) dx dy dz to an integral in terms of u, v, w involves replacing dx dy dz with:
A) |∂(x, y, z) / ∂(u, v, w)| du dv dw
B) ∂(x, y, z) / ∂(u, v, w) du dv dw
C) |∂(u, v, w) / ∂(x, y, z)| du dv dw
D) du dv dw
40. Consider the transformation x = u, y = uv. What is ∂(x, y) / ∂(u, v)?
A) u
B) v
C) uv
D) 1
41. For a transformation from (u, v, w) to (x, y, z), the Jacobian determinant is given by:
A) det(∂(x, y, z) / ∂(u, v, w))
B) det(∂(u, v, w) / ∂(x, y, z))
C) The sum of the partial derivatives.
D) The product of the partial derivatives.
42. Which of the following is NOT a typical use case for Jacobians in integral calculus?
A) Changing variables in multiple integrals.
B) Simplifying integration regions.
C) Calculating surface areas of complex shapes.
D) Evaluating line integrals along specific paths.
43. The Jacobian for the transformation from Cartesian coordinates (x, y, z) to spherical coordinates (ρ, θ, φ) is:
A) ρ^2 sin(φ)
B) ρ sin(φ)
C) ρ^2 cos(φ)
D) ρ
44. If we want to evaluate the integral ∫∫_R (x^2 + y^2) dx dy over a region R, and we use polar coordinates x = r cos(θ), y = r sin(θ), the integral becomes:
A) ∫∫_S r^2 * r dr dθ
B) ∫∫_S r^2 dr dθ
C) ∫∫_S r^3 dr dθ
D) ∫∫_S (r cos(θ))^2 + (r sin(θ))^2 dr dθ
45. Consider the transformation x = u + v, y = u - v. What is ∂(x, y) / ∂(u, v)?
A) 2
B) -2
C) 0
D) 1
46. What is the geometric interpretation of the absolute value of the Jacobian |∂(x, y) / ∂(u, v)|?
A) It represents the change in area under the transformation.
B) It represents the change in volume under the transformation.
C) It represents the change in arc length under the transformation.
D) It represents the change in surface area under the transformation.
47. The transformation of a double integral ∫∫_R f(x, y) dx dy to an integral in terms of u and v involves replacing dx dy with:
A) |∂(x, y) / ∂(u, v)| du dv
B) |∂(u, v) / ∂(x, y)| du dv
C) ∂(x, y) / ∂(u, v) du dv
D) du dv
48. If x = r cos(θ) and y = r sin(θ), what is the Jacobian of the transformation from (r, θ) to (x, y)?
A) r
B) r^2
C) 1/r
D) 0
49. The Jacobian of a transformation from (u, v) to (x, y) is denoted by:
A) ∂(x, y) / ∂(u, v)
B) ∂(u, v) / ∂(x, y)
C) J(u, v)
D) J(x, y)
50. What is a Jacobian in the context of multivariate calculus?
A) The determinant of the Hessian matrix of a scalar function.
B) The determinant of the matrix of partial derivatives of a vector function.
C) The sum of the second partial derivatives of a function.
D) The average value of a function over a region.
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