Mathematics 373 – Applied Mathematics (PDE) Individual Problem Set 1 – First Order Linear PDE January 18, 2001

A. (Review from Multivariable Calculus) Recall that the chain rule from 1-variable calculus says that if g is differentiable at x and f is differentiable at g(x) then the composition f(g(x)) is differentiable at x and

\frac{d}{dx}f(g(x)) = f′(g(x))g′(x)

A first generalization of this to functions of several variables is the following: If f(x, y) is a differentiable function of two variables and α(t) = (x(t), y(t)) is a differentiable parametric curve with image in the domain of f, then

(1)  \frac{d}{dt}f(α(t)) = \frac{∂f}{∂x} (x(t), y(t))x′(t) + \frac{∂f}{∂y} (x(t), y(t))y′(t) = ∇f(α(t)) · α′(t)

where ∇f is the gradient vector of f.

1) Use (1) to compute \frac{d}{dt}f(α(t)) if f(x, y) = x^{2} − y^{2} + x^{3} and α(t) = (t^{2} − 1, t^{3} − t). What does your result say about the curve α and f?

2) If α(t) = (x_{0}, y_{0}) + t(u, v) where (u, v) is a unit vector,

\frac{d}{dt}f(α(t))|_{t=0} = ∇f(α(0)) · α′(0) = ∇f(x_{0}, y_{0}) · (u, v)

is the directional derivative of f at α(0) = (x_{0}, y_{0}) in the direction of the vector α′(0) = (u, v). Fixing the point (x_{0}, y_{0}), consider the directional derivative as a function of the direction vector (u, v). In which direction is the directional derivative maximized? In which direction is the directional derivative minimized? In which directions is the directional derivative equal to zero? Prove your assertions.

3) The line integral of a vector field F(x, y) = (P (x, y), Q(x, y)) along a parametric curve α(t) = (x(t), y(t)) (a ≤ t ≤ b) in the plane is defined to be

∫_{α} F = ∫_{a}^{b} F(α(t)) · α′(t) dt

Show that if F = ∇f for some f(x, y), then the line integral doesn’t actually depend on the path α, only on the starting and final points of the path: α(a) and α(b).

B. Solve the PDE 4u_{x} + 7u_{t} = 0 with the Cauchy condition u(x, 0) = sin(3x) − cos(2x).

C. Use the change of coordinates method to solve u_{x} + 2u_{y} + u = e^{3x−2y} with u(x, 0) = 0.

D. Solve (1 + x^{2})u_{x} + u_{y} = 0 for u(x, y). Sketch some of the characteristic curves.

E. Solve yu_{x} + xu_{y} = 0 with u(0, y) = e^{−y^{2}}. In which region of the plane is the solution uniquely determined?

F. What happens if we try to solve a first order linear PDE a(x, y)u_{x} + b(x, y)u_{y} = 0 with the condition u(x, 0) = f(x), but the x-axis is a characteristic curve of the equation? (Note: There are two cases!)

Assignment

Individual write-ups due in class on Thursday, January 25.