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January 13th, 2012, 06:18 AM   #1
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Help with total differential

Given a function in the Cartesian plane f(x,y,z) with x, y and z being implicit functions of time
& from the definition of total derivative
dot f = df/dt = f_x dot x + f_y dot y + f_z dot z + f_t

where
f_t means partial derivative of f with t
dot f means df/dt

Now my question is it true that

dot x = f_y dot y + f_z dot z + x_t
and similarly
dot y = f_x dot x + f_z dot z + y_t
and
dot z = f_y dot y + f_x dot x + z_t

If these are not true then what is
dot x equal to?

Please help.
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January 13th, 2012, 06:59 AM   #2
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Re: Help of total derivative

This is wrong.

You can eventually state:



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January 16th, 2012, 03:45 AM   #3
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Re: Help of total derivative

Thanx wnvl for your answer:

But why is my original equation wrong? I would have thought that it is an application of the total derivative for x on a space with vectors x, y and z. In fact I was actually thinking of any random space where these three vectors maybe related but they are all implicitly related to time. So if I don't restrict x, y, and z to being the usual Cartesian coordinates what makes my original equation wrong.

I agree with your (wnvl) equation because it's a rearrangement of the total derivative, but what makes my equation wrong?

If I were to ask then


is the following true:

or is the following true:



or perhaps none are true?
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January 16th, 2012, 09:22 AM   #4
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Re: Help of total derivative

Quote:
Originally Posted by Taurai Mabhena


Problem with this equation I think is that you use the same notation x for a function x(x,y,z,t) and for the name of a variable in that function. I think you should use a different name for both.
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January 16th, 2012, 09:34 AM   #5
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If f is a function of x, y and z only (as in the original post), where x, y and z are functions of t and certain conditions are satisfied,

.

If you (separately) suppose that x is a function of y and z, where y and z are functions of t, and certain conditions are satisfied,

.
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January 16th, 2012, 02:03 PM   #6
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Re:

Quote:
Originally Posted by skipjack
If f is a function of x, y and z only (as in the original post), where x, y and z are functions of t and certain conditions are satisfied,

.

If you (separately) suppose that x is a function of y and z, where y and z are functions of t, and certain conditions are satisfied,

.
This is indeed what I meant. Here you are replacing one meaning of x by f.
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January 22nd, 2012, 11:26 PM   #7
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Re: Help of total derivative

Thanks skipjack and wnvl for elaborate answers. To skpjack what would be the conditions you mean when you say, "If you (separately) suppose that x is a function of y and z, where y and z are functions of t, and certain conditions are satisfied,"??
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January 22nd, 2012, 11:35 PM   #8
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Re: Help of total derivative

Further, am I to assume that skipjack's FULL answer is actually
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January 25th, 2012, 09:51 PM   #9
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Re: Conditions to be satisfied???

Quote:
Originally Posted by skipjack
If you (separately) suppose that x is a function of y and z, where y and z are functions of t, and certain conditions are satisfied,
.
Thanks skipjack and wnvl for elaborate answers. To skipjack or anyone who can answer, what would be the conditions you mean when you say, "If you (separately) suppose that x is a function of y and z, where y and z are functions of t, and certain conditions are satisfied,"??
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January 27th, 2012, 03:53 PM   #10
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Quote:
Originally Posted by Taurai Mabhena
am I to assume that skipjack's FULL answer . . .
No.

Quote:
Originally Posted by Taurai Mabhena
. . . what would be the conditions you mean . . . ?
Sufficient conditions are that the derivatives used (on the right-hand side) exist and are continuous where relevant.
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