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 January 13th, 2012, 06:18 AM #1 Newbie   Joined: Jan 2012 Posts: 6 Thanks: 0 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.
 January 13th, 2012, 06:59 AM #2 Senior Member   Joined: Oct 2011 From: Belgium Posts: 522 Thanks: 0 Re: Help of total derivative This is wrong. You can eventually state: $\dot{f(x(t),y(t),z(t),t)}=\frac{\partial f}{\partial x}\dot{x}+\frac{\partial f}{\partial y}\dot{y}+\frac{\partial f}{\partial z}\dot{z}+\frac{\partial f}{\partial t}$ $\dot{x}=\frac{\dot{f(x(t),y(t),z(t),t)}-\frac{\partial f}{\partial y}\dot{y}-\frac{\partial f}{\partial z}\dot{z}-\frac{\partial f}{\partial t}}{\frac{\partial f}{\partial x}}$
 January 16th, 2012, 03:45 AM #3 Newbie   Joined: Jan 2012 Posts: 6 Thanks: 0 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: $\begin{equation} {eqn1:} \dot{x} = \frac{dx}{dt}=\frac{\partial x}{ \partial x}\dot{x} + \frac{\partial x}{ \partial y}\dot{y} + \frac{\partial x}{ \partial z}\dot{z} + \frac{\partial x}{ \partial t} \end{equation}$ or is the following true: $\begin{equation} {eqn2:} \dot{x} = \frac{dx}{dt} = \frac{\partial x}{ \partial y}\dot{y} + \frac{\partial x}{ \partial z}\dot{z} + \frac{\partial x}{ \partial t} \end{equation}$ or perhaps none are true?
January 16th, 2012, 09:22 AM   #4
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Re: Help of total derivative

Quote:
 Originally Posted by Taurai Mabhena $\begin{equation} {eqn1:} \dot{x} = \frac{dx}{dt}=\frac{\partial x}{ \partial x}\dot{x} + \frac{\partial x}{ \partial y}\dot{y} + \frac{\partial x}{ \partial z}\dot{z} + \frac{\partial x}{ \partial t} \end{equation}$
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.

 January 16th, 2012, 09:34 AM #5 Global Moderator   Joined: Dec 2006 Posts: 21,035 Thanks: 2272 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, $\frac{df}{dt}\,=\,\frac{\partial f}{\partial x}\dot{x}\,+\,\frac{\partial f}{\partial y}\dot{y}\,+\,\frac{\partial f}{\partial z}\dot{z}$. 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, $\dot{x}\,=\,\frac{dx}{dt}\,=\,\frac{\partial x}{\partial y}\dot{y}\,+\,\frac{\partial x}{\partial z}\dot{z}$.
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, $\frac{df}{dt}\,=\,\frac{\partial f}{\partial x}\dot{x}\,+\,\frac{\partial f}{\partial y}\dot{y}\,+\,\frac{\partial f}{\partial z}\dot{z}$. 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, $\dot{x}\,=\,\frac{dx}{dt}\,=\,\frac{\partial x}{\partial y}\dot{y}\,+\,\frac{\partial x}{\partial z}\dot{z}$.
This is indeed what I meant. Here you are replacing one meaning of x by f.

 January 22nd, 2012, 11:26 PM #7 Newbie   Joined: Jan 2012 Posts: 6 Thanks: 0 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,"??
 January 22nd, 2012, 11:35 PM #8 Newbie   Joined: Jan 2012 Posts: 6 Thanks: 0 Re: Help of total derivative Further, am I to assume that skipjack's FULL answer is actually $\dot{x}= \frac{\partial x}{\partial y}\dot{y} + \frac{\partial x}{\partial z}\dot{z} + \frac{\partial x}{\partial t}$
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, $\dot{x}\,=\,\frac{dx}{dt}\,=\,\frac{\partial x}{\partial y}\dot{y}\,+\,\frac{\partial x}{\partial z}\dot{z}$.
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,"??

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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