need help with math1141 question (1 Viewer)

[maths > english]

A function f(x,y) is said to be homogeneous of degree n if f(tx,ty) = tⁿf(x,y) for all t>0. Show that such a function satisfies the equation:

x * ^∂f/_∂x + y * ^∂f/_∂y = nf

thanks in advance to anyone that can help

 

[chris_c28]

A function f(x,y) is said to be homogeneous of degree n if f(tx,ty) = tⁿf(x,y) for all t>0. Show that such a function satisfies the equation:

x * ^∂f/_∂x + y * ^∂f/_∂y = nf

thanks in advance to anyone that can help

Here's how I understand the problem

let Φ = f (tx,ty) = t^n f(x,y)
dΦ/dt = (∂f/∂x)*(dx/dt) + (∂f/∂y)*(dy/dt) = n*t^(n-1) f(x,y)
this gives
dΦ/dt = (∂f/∂x)*x + (∂f/∂y)*y = n*t^(n-1) f(x,y)

to show this equals the proposition, substitute t = 1

(∂f/∂x)*x + (∂f/∂y)*y = n*f(x,y)

If equation is true for t = 1 which is > 0, it should also be true for any t > 1

I'm worried bout the exams too. Seriously screwing up Discrete.

 

[maths > english]

thanks again for ur help

im screwing up discrete too, hate that subject

 

[redslert]

Discrete is FUN FUN FUN

 

[mojako]

Here's how I understand the problem

let Φ = f (tx,ty) = t^n f(x,y)
dΦ/dt = (∂f/∂x)*(dx/dt) + (∂f/∂y)*(dy/dt) = n*t^(n-1) f(x,y)
this gives
dΦ/dt = (∂f/∂x)*x + (∂f/∂y)*y = n*t^(n-1) f(x,y)

to show this equals the proposition, substitute t = 1

(∂f/∂x)*x + (∂f/∂y)*y = n*f(x,y)

If equation is true for t = 1 which is > 0, it should also be true for any t > 1

I'm worried bout the exams too. Seriously screwing up Discrete.

u dont sub t=1

let Φ = f (tx,ty) = t^n f(x,y)
dΦ/dt = (∂f/∂tx)*(dtx/dt) + (∂f/∂ty)*(dty/dt) = n*t^(n-1) f(x,y)
(∂f/∂tx)*x + (∂f/∂ty)*y = n*t^(n-1) f(x,y)
(∂f/∂tx)*tx + (∂f/∂ty)*ty = n*t^n f(x,y)
(∂f/∂tx)*tx + (∂f/∂ty)*ty = n*f(tx,ty)
(∂f/∂u)*u + (∂f/∂v)*v = n*f(u,v)
(∂f/∂x)*x + (∂f/∂y)*y = n*f(x,y)

btw do u know how to do Q43 c-e in last chapter of the algebra booklet?

 

[wogboy]

c)

J.x = (mx X v).x (X = cross product)
= m(x X v).x
= 0 (since x X v is orthogonal to x)

The locus of x is described by J.(x - 0) = 0, which describes a plane through the origin, with normal J

d)
J = (mx X v)
|J| = m|(x X v)|
= m|x||v| (since x and v are orthogonal in circular motion)
->|v| = |J||x|/m
J is a constant (from part a)
Since the particle is moving in a circle, |x| (the radius) is constant.
So the speed |v| is constant

e)
The infinitesimal area dA swept out in time dt by the radius vector x, moving at velocity v is dA = |x X v|/2 dt
-> dA/dt = |x X v|/2
= |J|/2m (which is constant)