# Partial differentiationWatch

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#1
if anyone can give me a hand with these partial differentiation qus would really be appreciated:

1)if f(x,y) is a scalar function of position defined on the x-y plane. The position of a fixed point may also be specified by cartesian coordinates u,v defined by axes that are rotated by an angle theta frin the x and y axes. Show that:

[email protected] + [email protected] and v= [email protected] + [email protected]

deduce that:

((d^2)f)/(d(x^2)) + ((d^2)f)/(d(y^2) = ((d^2)f)/(d(u^2)) + ((d^2)f)/(d(v^2))

i.e the 2 dimensional (triangle)^2 operator is invariant under a rotation of the axes.

(if someone can please explain in english what this all means???)

2) the speed v of a 1D river depends upon time t and position x. a log drifts freely in the river so that its speed matches the local speed of the river. the log is at position x(t) at time t. find the log's acceleration at time t in each of the cases

i) v=t^2
ii)v= x^2
iii) v=xt

give a general expression for this acceleration in terms of v and its partial derivatives.

(this one makes slightly more sense but still slightly confused)

thankyou xx
0
14 years ago
#2
First off this is definately not in the right forum. Partials aren't covered in ALevel. It also seems like you are from an institution of higher education that I am not particularly fond of.
However the first question looks relatively easy. Just find the values of each of the terms using the partial chain rule in the form df/du = df/dx dx/du + df/dy dy/du (where all d's are curly d's for partial).
0
14 years ago
#3
(Original post by Mehh)
First off this is definately not in the right forum. Partials aren't covered in ALevel. It also seems like you are from an institution of higher education that I am not particularly fond of.
However the first question looks relatively easy. Just find the values of each of the terms using the partial chain rule in the form df/du = df/dx dx/du + df/dy dy/du (where all d's are curly d's for partial).
LoL... first off this is deff the right forum... Looks like a mechanics question to me. And this is a maths forum

Is partial differentiation the same as implicit differentiation?
0
14 years ago
#4
No partial differentiation treats all variables that are not being differentiated as constant. And by (triangle) do you mean nabla?
0
#5
(Original post by AntiMagicMan)
No partial differentiation treats all variables that are not being differentiated as constant. And by (triangle) do you mean nabla?

could be nabla, is a triangle with the point downwards but don't know what its called
0
14 years ago
#6
(Original post by devilschild)
1)if f(x,y) is a scalar function of position defined on the x-y plane. The position of a fixed point may also be specified by cartesian coordinates u,v defined by axes that are rotated by an angle theta frin the x and y axes. Show that:

[email protected] + [email protected] and v= [email protected] + [email protected]

deduce that:

((d^2)f)/(d(x^2)) + ((d^2)f)/(d(y^2) = ((d^2)f)/(d(u^2)) + ((d^2)f)/(d(v^2))

i.e the 2 dimensional (triangle)^2 operator is invariant under a rotation of the axes.

(if someone can please explain in english what this all means???)
Firstly, to explain the last comment, the u and v co-ordinates are related to x and y by a rotation matrix. So we're looking to see if the Laplacian is invariant under a rotation.

Now (with d denoting partial differentiation) by the chain rule

df/dx = df/du du/dx + df/dv dv/dx

= df/du [email protected] - df/dv [email protected]

Differentiating again

d^2f/dx^2 =

[email protected] (d^2f/du^2 du/dx + d^2f/dudv dv/dx)
- [email protected] (d^2f/dudv du/dx + d^2f/dv^2 dv/dx)

= cos^[email protected] d^2f/du^2 - [email protected]@ d^2f/du/dv + sin^[email protected] d^2f/dv^2

0
14 years ago
#7
(Original post by Syncman)
LoL... first off this is deff the right forum... Looks like a mechanics question to me. And this is a maths forum

Is partial differentiation the same as implicit differentiation?
My referance was towards the fact that partials are covered in degree level...this is not the degree level forums, I think you will find.
One would have expected you to have been able to have seen some kind of a difference when I used terms like curly d....
0
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