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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: 
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: or is the following true: or perhaps none are true? 
January 16th, 2012, 09:22 AM  #4  
Senior Member Joined: Oct 2011 From: Belgium Posts: 522 Thanks: 0  Re: Help of total derivative Quote:
 
January 16th, 2012, 09:34 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,805 Thanks: 2150 
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, . 
January 16th, 2012, 02:03 PM  #6  
Senior Member Joined: Oct 2011 From: Belgium Posts: 522 Thanks: 0  Re: Quote:
 
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 
January 25th, 2012, 09:51 PM  #9  
Newbie Joined: Jan 2012 Posts: 6 Thanks: 0  Re: Conditions to be satisfied??? Quote:
 
January 27th, 2012, 03:53 PM  #10  
Global Moderator Joined: Dec 2006 Posts: 20,805 Thanks: 2150  Quote:
Quote:
 

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