# practice exam

## Transcription

practice exam

Math 425 April 2015 Advanced Calculus Robin Young Practice Final Exam Name: 1. Show that Z `Z t Z 0 0 ` (` − x) f (x) dx. f (x) dx dt = 0 2. The following system arises in the study of gases: u − v = (h − α) e−s w − u = (k − β) e−t α es = β et . (a) Under what circumstances can we solve for α, β and u in terms of the remaining variables? ∂ ∂ ∂ (b) Calculate ∂s u, ∂s α and ∂s β by exhibiting a linear set of equations for them. No need to solve the system! 3. Calculate the volume and surface area of the unit sphere. 4. Recall that the work done by the force F along the curve Γ is Z Z W = F · dx = F · T ds. Γ Γ 2 Use Newton’s law F = m a = m ddt2x to show that the work done is the change of kinetic energy, 1 1 W = m v2 − m v02 . 2 2 5. Calculate Z x dy dz + y dx dz − y dx dy, S where S is the boundary of the solid cylinder x2 + y 2 ≤ 1, 0 ≤ z ≤ 1. 6. (a) Show that the 1-form ω = (3x2 y 4 − 2x) dx + (4x3 y 3 + 3y 2 ) dy + cosh z dz is closed. (b) By integrating along an appropriate curve, show that ω is exact: that is, find a function (0-form) α such that ω = dα. 7. If F(x, y, z) is a vector field which is differentiable everywhere, show that Z Z (∇ × F) · dS S vanishes for every closed surface S (i.e. S encloses some volume). 8. Let r = (x1 , . . . , xn ) ∈ Rn and let r = krk be the radial distance function. 2 (a) Calculate the quantities ∇ r and ∇ · r. (b) If f˜(x1 , . . . , xn ) is a radial function, f˜(x1 , . . . , xn ) = f (r) for a scalar function f , show that its Laplacian ∆f˜ ≡ ∇ · ∇ f˜ is given by ∆f˜ = f 00 (r) + n−1 0 f (r). r 9. Under what circumstances can the pair of equations x2 − y 2 − u3 + v 2 = −4 2 2 2xy + y − 2u + 3v 4 = −8 be solved to give (u, v) in terms of (x, y)? ∂ u at the point (x, y, u, v) = (2, −1, 2, 1). Find ∂x 10. A 1-form ω is radial if it is of the form ω = f (r) (x dx + y dy + z dz) , p where f is a scalar function and r = x2 + y 2 + z 2 . (a) Show that a radial 1-form satisfies ω = 1 2 f (r) d(r2 ). (b) Show that if ω is radial, then its exterior derivative vanishes, d ω = 0. (c) Find some function g which is a scalar potential for ω, i.e. for which ω = dg. 3 (1) (2)