# What is Total derivative: Definition + 48 Threads

In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when f is a function of several variables, because when f is a function of a single variable, the total derivative is the same as the derivative of the function."Total derivative" is sometimes also used as a synonym for the material derivative in fluid mechanics.

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1. ### Line element in cylindrical coordinates

First I took the total derivative of these and arrived at $$dr=\frac{\partial r}{\partial x}dx+\frac{\partial r}{\partial y}dy \quad\rightarrow \quad r²dr=xdx+ydy$$ $$d\phi=\frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy \quad\rightarrow \quad r²dr \phi=-ydx+xdy$$...
2. ### I 4d integration/differentiation notation and the total derivative

This is probably a stupid question but, ## \frac{d\partial_p}{d\partial_c}=\delta^p_c ## For the notation of a 4D integral it is ##d^4x=dx^{\nu}##, so if I consider a total derivative: ##\int\limits^{x_f}_{x_i} \partial_{\mu} (\phi) d^4 x = \phi \mid^{x_f}_{x_i} ## why is there no...
3. ### Calculating total derivative of multivariable function

This isn't a homework problem exactly but my attempt to derive a result given in a textbook for myself. Below is my attempt at a solution, typed up elsewhere with nice formatting so didn't want to redo it all. Direct image link here. Would greatly appreciate if anyone has any pointers.
4. ### Find the total derivative of ##u## with respect to ##x##

see attached below; the textbook i have has many errors... clearly ##f_x## is wrong messing up the whole working to solution...we ought to have; ##\frac {du}{dx}=(9x^2+2y)+(2x+8y)3=9x^2+2y+6x+24y=9x^2+6x+26y##
5. ### Calculating 2nd Total Derivative of u w.r.t. t

du/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt) So i write the operator as d/dt = (dx/dt)(∂/∂x) + (dy/dt)(∂/∂y) and apply it to du/dt ; in the operator it is the partial derivative that acts on du/dt which involves using the product rule. I am having a problem with the term involving (∂/∂x)...
6. ### A Partial / Total Derivative, Compositions

Hello there, I have stumbled across further examples to derivatives of multivariable functions that confuse me. Similar to my other thread: https://www.physicsforums.com/threads/partial-derivative-of-composition.985371/#post-6309196 Suppose we have two functions, ## f: R^2 \rightarrow R...
7. ### Commutativity of partial and total derivative

Problem Statement: Use the definition of the total time derivative to a) show that ##(∂ /∂q)(d/dt)f(q,q˙,t) = (d /dt)(∂/∂q)f(q,q˙,t)## i.e. these derivatives commute for any function ##f = f(q, q˙,t)##. Relevant Equations: My approach is given below. Please tell if it is correct and if not ...
8. ### I What is the 'formal' definition for Total Derivative?

A total derivative dU = (dU/dx)dx + (dU/dy)dy + (dU/dz)dz. I am unsure of how to use latex in the text boxes; so the terms in parenthesis should describe partial differentiations. My question is, where does this equation comes from?
9. ### I Total Derivative of a Constrained System

Hi all, I was working on a problem using Euler-Lagrange equations, and I started wondering about the total and partial derivatives. After some fiddling around in equations, I feel like I have confused myself a bit. I'm not a mathematician by training, so there must exist some terminology which...
10. ### I The total derivative of a wavefunction

I have read that the integral of d3x ∇(ψ*ψ) is zero because the total derivative vanishes if ψ is normalizable. Does this mean that the integral of d3x ∇(ψ*ψ) is ψ*ψ evaluated at the limits where ψ is zero ? Thanks
11. ### I What are the insights into the Total Derivative formula?

I’ve always been confused by the formula for the Total Derivative of a function. $$\frac{df(u,v)}{dx}= \frac{\partial f}{\partial x}+\frac{\partial f }{\partial u}\frac{\mathrm{d}u }{\mathrm{d} x}+\frac{\partial f}{\partial v}\frac{\mathrm{d}v }{\mathrm{d} x}$$ Any insight would be greatly...
12. ### Derive lowest order (linear) approximation

Homework Statement For a single mechanical unit lung, assume that the relationship among pressure, volume, and number of moles of ideal gas in the ling is given by PA((VL)/(NL)a = K, where a = 1 and K is a constant. Derive the lowest-order (linear approximation to the relationship among changes...
13. ### I Show How Theta Term in QCD Lagrangian is a Total Derivative

I'm trying to show that the theta term in the QCD Lagrangian, ##\alpha G^a_{\mu\nu} \widetilde{G^a_{\mu\nu}}##, can be written as a total derivative, where ##$$G^a_{\mu\nu} = \partial_{\mu} G^a_{\nu} - \partial_{\nu}G^a_{\mu}-gf_{bca}G^b_{\mu}G^c_{\nu}$$ ##...

22. ### Total derivative of a partial derivative

Im doing a question on functionals and I have to use the Euler lagrange equation for a single function with a second derivative. My problem is I don't know how to evaluate \frac{d^2}{dx^2}(\frac{\partial F}{\partial y''}). Here y is a function of x, so y'=\frac{dy}{dx}. I know this is probably...
23. ### How Does the Total Derivative Sum Up Changes in Multiple Directions?

##dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy## I'm confused as to how the total derivative represents the total change in a function. My own interpretation, which I know is incorrect, is that ##\frac{\partial z}{\partial x} dx## represents change in the x...
24. ### Total derivative and partial derivative

can anyone tell me the difference of application of total derivative and partial derivative in physics? i still can't figure it out after searching on the internet
25. ### Difference of total derivative and partial derivative

many books only tell the operation of total derivative and partial derivative, so i now confuse the application of these two. when doing problem, when should i use total derivative and when should i use partial derivative. such a difference is detrimental when doing Physics problem, so i...
26. ### Adding a total derivative to the Lagrangian

I recently posted another thread on the General Physics sub forum, but didn't get as much feedback as I was hoping for, regarding this issue. Let's say I have two Lagrangians: $$\mathcal{L}_1 = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\mu A^\mu)^2$$ ...
27. ### Total derivative with a constraint

Hi there, I have what I suspect is a straightforward question. I wish to take the total derivative of the following function: W(q,x) = q \cdot u(x) + c(q,x) Subject to the constraint: \frac{q}{x}=\bar{m}, where \bar{m} is some constant > 0, and c(q,x) is additively separable...
28. ### Euler characteristic as a total derivative

We all know that the Euler characteristic is a topological invariant. But let's suppose that we don't know this or anything else about algebraic topology for that matter. We are given only the Gauss-Bonnet theorem, which expresses the Euler characteristic in geometrical terms. In his string...
29. ### Total derivative to partial derivative by division? (Calc./Thermo.)

I don't understand the calculus behind this thermodynamics concept: S = f(T,P) dS = (∂S/∂T)_P*dT + (∂S/∂P)_T*dP (∂S/∂T)_V = (∂S/∂T)_P + (∂S/∂P)_T*(∂P/∂T)_V Basically, I don't get why and how you get (∂S/∂T) when you divide dS by dT. Also, I don't understand why the constant volume...
30. ### Motivation to Create the Total Derivative

If I have w(x, y, z) and take "dw = (∂w/∂x)dx + (∂w/∂y)dy + (∂w/∂z)dz"; doesn't that accurately describe how w changes as x, y and z change by an infinitesimal? Or does that only work for some special cases? I feel like if I take the total derivative I am actually describing how the function w...
31. ### What exactly is a 'total derivative'?

Homework Statement What exactly is a total derivative? What is the definition of this concept? Homework Equations An example of total derivatives: The Attempt at a Solution I've tried searching for it, but found no helpful information.
32. ### Total derivative of integral seen as a functional, how?

To be specific, with total derivative I mean the linear map that best approximates a given function f at a given point. For f:ℝ\toℝ we have D(f,x_0):ℝ\toℝ, i.e. D(f,x_0)(h) \in ℝ. Often it is also denoted as just \delta f. Now in physics, in particular in the area of the Lagrangian, I find...
33. ### Error computing total derivative

find the total derivative dz/dt, given: z=(x^2)-8xy-(y^3) where x=3t and y=1-t my steps look like this, can someone point out where i am going wrong, please? z'=2x-8(x'y+y'x)-3y^2 where x'=3 and y'=-1 2(3t)-8[3(1-t)+(-1)(3t)]-3(1-t)^2 =6t-8[3-3t+(-3t)]-3(1-2t+t^2) =6t-8(3-6t)-3(1-2t+t^2)...
34. ### Dividing two differentials gives a total derivative how?

Hello I am studying some differential geometry. I think I have understood the meaning of "differential" of a function: \text{d}f (V) = V(f) It is a 1-form, an operator that takes a vector and outputs a real number. But how is it related to the operation of "total derivative" ? For...
35. ### Exploring the Total Derivative of Gibbs Free Energy in a Closed System"

Hi everyone Homework Statement Let's say I want to do the totale drivative of the Gibbs free energy in dependent of: volume, temperature, amount of substance and surface. And let's say afterwards we have a closed system where the temperature is constant. How does the total derivative...
36. ### Is Equation a Total Derivative?

Homework Statement Is the following equation a total derivative? dz = 2ln(y)dx+{\frac x y}dy Homework Equations - The Attempt at a Solution I would say no. I tried it with the symmetry of the second derivatives. 2ln(y) is {\frac {\partial z} {\partial x}} when I...
37. ### The total derivative and a line segment S

Homework Statement Let f be a differentiable function from an open set U\subseteqR^{n} into R. If x,y\inU and the segment S={(1-t)x+yt : t\in[0,1]} is contained in U, show that f(y)-f(x)=(Df)_{\xi}(y-x) for some \xi\inS. The Attempt at a Solution The only direction I have...
38. ### Total Derivative of a Function of Two Variables

greetings, consider a function f(x,y); the total derivative of a function of two varible is given by-: df=(dou)f/(dou)x*dx+(dou)f/(dou)y*dy here we have the differential of f(x,y).but i am not able to understand why the term dx and dy has appeared? advanced thanks.
39. ### Multivariable analog to the total derivative?

For a single variable we have \int_{x_1}^{x_2} f(x) dx = F(x_2)-F(x_1) if f(x) = dF/dx. f(x) is then a total derivative. What is the analog in 3D so that \int_V f(\vec{x}) d^3x does not depend on the values of f in the interior of V? In case there is not a single answer...
40. ### Total derivative: elementary question

Hello, Given a function f(x_1,\ldots,x_n), the total derivative for x_1 is: \frac{df}{dx_1}=\frac{\partial f}{\partial x_1}+\frac{\partial f}{\partial x_2}\frac{dx_2}{dx_1}+\ldots+\frac{\partial f}{\partial x_n}\frac{dx_n}{dx_1} Now, if the x_i are just variables, can we say that...
41. ### Proving Total Derivative of jf+kg at a

Homework Statement If U⊆R^n is an open set with a ∈ U, and f: U->R^m and g: U->R^m are totally differentiable at a, prove that jf+kg is also totally differentiable at a and that (D(jf+kg))a = j(Df)a+k(Dg)a. Homework Equations The Attempt at a Solution Let p(x) = jf(x)+kg(x) Then...
42. ### Understanding the total derivative in terms of analysis

I'm trying to work through an introduction to Lagrangian mechanics. I get the idea. You have a particle at point A traveling to point B. You have a functional which maps every path between those points to a scalar called the action of the path. The universe then does some number crunching...
43. ### Understanding Total Derivative: Step-By-Step Guide

Hi I was hoping someone could explain how this works to me I ran across this example in a book D(p(a)) = S(p(a),a) Total derivative with respect a \frac{dD(p(a))}{dp} \frac{dp}{da}= \frac{\partial S(p(a),a)}{\partial p} \frac{dp}{da} + \frac{\partial S(p(a),a)}{\partial a}
44. ### Total Derivative in Curvilinear Coordinates

I'm trying to study some basic tensor analysis on my own for practical purposes, but I'm having some problems. More specifically I'm rather puzzled over the concept of total derivative in curvilinear coordinates (well, to be exact, as I've got little experience with differential geometry, it's...
45. ### Total derivative formula nonsense

Total derivative formula confusion It took me over an hour to fully resolve the confusion that appears in textbooks about the total derivate formula. Some textbooks use the term total derivative if a function f is a function t and other variables, and each of those variables themselves are...
46. ### Replacing total derivative with partial derivative in Griffiths' book

I'm using Griffiths' book to self-study QM and I'm having a slight problem following one of his equations. In page 11 of his "Intro to Quantum Mechanics (2nd ed.)", he gives the reader the following 2 equations: \frac {d} {dt} \int_{-\infty}^{\infty}|\Psi(x,t)|^2 dx = \int_{-\infty}^{\infty}...
47. ### Partial derivatives/ total derivative

Homework Statement I've attactched an image of the question, I hope this is ok, if not let me know and I'll copy it out onto a post, The Attempt at a Solution I've done parts (a) and (b) using the total derivative of f ( http://mathworld.wolfram.com/TotalDerivative.html ) but I can't get...
48. ### Total derivative -> partial derivative

Under what conditions can you replace a total differential with a partial? dx/dy -> partial(dx/dy) in the context of 2 independant variables and multiple dependant variables. Thanks