Calculus IV Homework: Partial Derivatives, Wave Equation, Harmonic Functions, Acceleration | Assignments Advanced Calculus

Math 2242 (Calculus IV) Graded Homework 2

DUE 02/26/09

1. (Partial Derivatives) (10 points each):

(a) Let f:R2→R: (x, y)7→ exsin2(cos(xy)) x4y3−143πcos(exsin y). Does

∂2f

∂x∂y =∂2f

∂y∂ x ?

[Hint: Do NOT try and calculate the partial derivatives. Cite an appropriate theorem instead.]

(b) Let fand gbe C2functions of one variable. Set ϕ=f(x−t) + g(x+t). Prove that ϕsatisfies

the wave equation

∂2ϕ

∂t2=∂2ϕ

∂x2.

[For example, if f(z) = esinzand g(z) = z2then f(x−t) + g(x+t) = esin(x−t)+ (x+t)2satisfies

the wave equation. Obviously wave equations are important in science and engineering.]

∂2u

∂x2+∂2u

∂y2= 0

is called a harmonic function. Show that the function u(x, y) = x3−3xy2is harmonic. [Har-

monic functions are extremely important in science, engineering, complex analysis, and even

number theory!]

2. (Acceleration and Newton’s Second Law) (10 points): Suppose an asteroid with mass mmoves along

the ellipse-like path r(t) = (3 cos t,2 sin t) for 0 ≤t≤2πabout the sun. According to Newton’s second

law, F=ma=mr00(t). Find the centripetal force acting on the asteroid at time t=π

2, i.e. F(π

2).

[The path given in this problem is not really an ellipse, so by Kepler’s first law, the asteroid would not

actually move like this. But never mind that. For fun, explain why the direction of the force vector

you found makes sense.]

3. (Arc Length) (10 points): Write down the integral that gives the arc length of one orbit of the asteroid

in Problem 2. Then use your calculator (or computer) to give a decimal approximation.

4. (Vector Fields and Flow Lines) (10 points each):

(a) Sketch the vector field F= (−x,y ). [Hint: Draw some vectors on the coordinate axes first.]

(b) Draw a few flow lines on your sketch.

field passing through the point (x0, y0) at time t= 0. Then check your work by verifying that

c0(t) = F(c(t)). [Hint: If c(t)=(x(t), y(t)) is such a flow line, then it must satisfy the following

system of ordinary differential equations

dt=−x

dt=y,

subject to the initial conditions x(0) = x0and y(0) = y0.]

5. (Divergence, Curl, Laplacian) (10 points each):

(a) Let F= (cos(xy), exz ,sinh(z)). Calculate ∇ · F, i.e. the divergence of the vector field F. [Yes,

that is sinh (hyperbolic sine), not sin, in the z-component.] Is Fsolenoidal?

(b) Let F=yzˆ

i+xzˆ

j+xyˆ

k. Calculate ∇ × F, i.e the curl of the vector field F. Is Firrotational?

[Hint: ∇2ϕ=∇·∇ϕ.]

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Math 2242 (Calculus IV) Graded Homework 2 DUE 02/26/

(Partial Derivatives) (10 points each):

(a) Let f : R^2 → R : (x, y) 7 → ex^ sin

(^2) (cos(xy)) x^4 y^3 − 143 π cos(ex^ sin^ y^ ). Does ∂^2 f ∂x∂y

∂^2 f ∂y∂x

[Hint: Do NOT try and calculate the partial derivatives. Cite an appropriate theorem instead.] (b) Let f and g be C^2 functions of one variable. Set ϕ = f (x − t) + g(x + t). Prove that ϕ satisfies the wave equation ∂^2 ϕ ∂t^2

∂^2 ϕ ∂x^2

[For example, if f (z) = esin^ z^ and g(z) = z^2 then f (x − t) + g(x + t) = esin(x−t)^ + (x + t)^2 satisfies the wave equation. Obviously wave equations are important in science and engineering.] (c) A function u(x, y) with continuous second partials (i.e. C^2 ) satisfying Laplace’s equation ∂^2 u ∂x^2

∂^2 u ∂y^2

is called a harmonic function. Show that the function u(x, y) = x^3 − 3 xy^2 is harmonic. [Har- monic functions are extremely important in science, engineering, complex analysis, and even number theory!]

(Acceleration and Newton’s Second Law) (10 points): Suppose an asteroid with mass m moves along the ellipse-like path r(t) = (3 cos t, 2 sin t) for 0 ≤ t ≤ 2 π about the sun. According to Newton’s second law, F = ma = mr′′(t). Find the centripetal force acting on the asteroid at time t = π 2 , i.e. F( π 2 ). [The path given in this problem is not really an ellipse, so by Kepler’s first law, the asteroid would not actually move like this. But never mind that. For fun, explain why the direction of the force vector you found makes sense.]
(Arc Length) (10 points): Write down the integral that gives the arc length of one orbit of the asteroid in Problem 2. Then use your calculator (or computer) to give a decimal approximation.
(Vector Fields and Flow Lines) (10 points each):

(a) Sketch the vector field F = (−x, y). [Hint: Draw some vectors on the coordinate axes first.] (b) Draw a few flow lines on your sketch. (c) (Bonus 10 points): For any point (x 0 , y 0 ), find a path c(t) that is a flow line of the vector field passing through the point (x 0 , y 0 ) at time t = 0. Then check your work by verifying that c′(t) = F(c(t)). [Hint: If c(t) = (x(t), y(t)) is such a flow line, then it must satisfy the following system of ordinary differential equations dx dt

= −x dy dt = y,

subject to the initial conditions x(0) = x 0 and y(0) = y 0 .]

(Divergence, Curl, Laplacian) (10 points each):

(a) Let F = (cos(xy), exz^ , sinh(z)). Calculate ∇ · F, i.e. the divergence of the vector field F. [Yes, that is sinh (hyperbolic sine), not sin, in the z-component.] Is F solenoidal? (b) Let F = yzˆi + xzˆj + xykˆ. Calculate ∇ × F, i.e the curl of the vector field F. Is F irrotational? (c) Let ϕ(x, y, z) = 3x^2 − sin(xy) + z. Calculate ∇^2 ϕ, i.e. the Laplacian of the scalar function ϕ. [Hint: ∇^2 ϕ = ∇ · ∇ϕ.]

Calculus IV Homework: Partial Derivatives, Wave Equation, Harmonic Functions, Acceleration, Assignments of Advanced Calculus

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