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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
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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. 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, . 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
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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. 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
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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,"?? 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. Tags differential, total Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Ines White Calculus 1 October 30th, 2013 07:43 AM branny12000 Advanced Statistics 0 May 28th, 2013 04:37 AM piotrek Differential Equations 2 May 23rd, 2013 07:22 AM reemas Calculus 3 October 22nd, 2011 07:21 PM MathematicallyObtuse Algebra 3 February 8th, 2011 01:41 AM

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