Differentials Definition and 236 Threads

My first ever post here, so hello world! So I've attached an image of a conventional 4x4 system with one RIC engine and LSDs for all wheels so that all four wheels can be driven, or just one depending on conditions. the front/rear LSD may in fact be locked, but that isn't particularly...

I am currently reading "Differential Equatons with Applications" by Ritger and Rose, and I need some clarification about some notation and convention that they are using. I think it all stems from a lack of clarity of the difference between the operator d/dx and the "object" (I don't know what...

Hi all, first post here. I'm a junior Physics/Math double major at UMass Amherst, playing with some problems over the summer. I'll get right into it. A rope with constant tension T is deflected through the angle 2\theta_{0} by a smooth, fixed pulley. What is the force on the pulley? It is...

Homework Statement The circumference of a sphere was measured to be 90 cm with a possible error of 0.5 cm. Use differentials to estimate the maximum error in the calculated volume. Homework Equations Volume of sphere: V=4/3πR3 Circumference of Sphere: C=2πR ΔC = 0.5 cm The Attempt at a...

This is probably an elementary question, but I stumbled upon it while thinking about total differentials. One of their many applications is calculating the error in a volume, for example, given uncertainties in its dimensions. I'm not in the mood to tackle a 3D problem, so let's revert to a 2D...

I have recently come across the use of differentials in visualizing and thinking about calculus. In this method, one thinks of dx/dy as an actual fraction of infinity small yet real numbers. How is it possible to apply this to implicit functions?

"Expanding the taylor series for ##f(x)##.." (See picture) is this a typo? Aren't we expanding ##f(x + \Delta x)##? Also, when we evaluate ##f(x)## (coefficients in the expansion), are we assuming ##\Delta x = 0## by setting ##x + \Delta x## (argument of the function) equal to ##x##? Or are we...

Hi everyone, Sorry if this has been posted before, but I had a quick question manipulating differentials. This problem is in the context of thermodynamics. We know from the first law that E=E(S,V), and from calculus I know that: dE=(∂E/∂S)v dS+(∂E/∂V)s dV sorry if this is hard to read, I'm new...

I had a couple of questions. 1. Why does the integral ∫exf(t) dt transform to ex∫f(t) dt? Shouldn't ex be a part of the integrand too? 2. Why is the difference dy - dy1 = d(y - y1)?

When I first came across differentials, I was told that they could be thought of as infinitesimal changes. However, I can't get my head around how they're actually used to model physical problems. For example, if ##x## is the x-coordinate of a moving body, then ##dx## is an infinitesimally small...

Consider the following two calculations: (1) d(x\cos x)=(\cos x-x\sin x)dx (2) \frac{d}{dx}(x\cos x)=(\cos x-x\sin x) I would describe these both as differentiation. Is there a standard terminology that allows one to make the distinction between the two, if desired? The best I could come up...

Suppose that we have this metric and want to find null paths: ds^2=-dt^2+dx^2 We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$ dx=\pm dt \to x=\pm t Now switch to the more abstract and rigorous one-forms in differentiable manifolds. Here \mathrm{d}t (v)...

Suppose that we have this metric and want to find null paths: ds^2=-dt^2+dx^2 We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$ dx=\pm dt \to x=\pm t Now switch to the more abstract and rigorous one-forms in differentiable manifolds. Here \mathrm{d}t (v)...

Disclaimer: This isn't a homework assignment, so maybe it shouldn't be in the homework forums. If you feel it should be located elsewhere, feel free to move it, but the template doesn't really apply to this question so... * * *...

Hi all, Went to a seminar today, arrived a few minutes late; hope someone can tell me something about this topic and/or give a ref so that I can read on it . I know this is a lot of material; if you can refer me to at least some if, I would appreciate it : 1)Basically, understanding how/why the...

Hey all, I just started a fluid mechanics class and I'm having trouble interpreting the physical meaning behind differentials in this free body diagram. For example, γδxδyδz. I know gamma is the specific weight of the block of fluid. And I know δ is the differential length in x, y, or z...

I was playing around with some simple differential equations earlier and I discovered something cool (at least for me). Suppose you have y=sin(x^2) \Rightarrow \frac{dy}{dx}=2xcos(x^2) What if, instead of taking the derivative with respect to x, I want to take the derivative with respect to...

Homework Statement If xs^2 + yt^2 = 1 and x^2s + y^2t = xy - 4 , find \frac{\partial x}{\partial s}, \frac{\partial x}{\partial t}, \frac{\partial y}{\partial s}, \frac{\partial y}{\partial t} , at (x, y, s, t) = (1, -3, 2, -1) Homework Equations The Attempt at a Solution I...

Hello! I'm reading Mary Boa's "mathematical methods in the physical sciences" and I'm on a section about total differentials. So a total differential is for f(x, y) we know to be: df = \frac{\partial f}{\partial x}{dx} + \frac{\partial f}{\partial y}{dy} Now, I've attached a...

Hey guys, I need some more help for this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. This is only for question 2. Ignore 1. Question: Alright, so from drawing a diagram, we know that width is "L" and length is "3L." Moreover, the area of...

Hi guys, first time posting here, but I have a question that I have been thinking about for quite a while, and I hope someone can help out with it. Assume a line of charge (with overall charge of +Q and of length L) that is lying on the x-axis. You want to calculate the electric field strength...

There exist exact differential for P (Pressure) ,V (volume) ,T (temperature), U (Internal Energy) but not for W(work), Q (heat) . Why?

I was reading a chapter on differentials in my calculus book, when I came across the graph shown in the image attached to this post. Two questions came to my mind upon seeing this graph: 1) Isn't it technically wrong to label the x-coordinates as x and (x + Δx)? I mean, wouldn't it be more...

Homework Statement A force of 500N is measured with a possible error of 1N. Its component in a direction 60° away from its line of action is required, where the angle is subject to an error of 0.5°. What (approximately) is the largest possible error in the component? Homework Equations...

please tell me if i did this correctly: task: I'm trying to divide the differential dA by dV where.. dA = differential surface area of a sphere, dV = differential volume of a sphere dA=8\pirdr dV=4\pir2dr so then dA/dV= 2/r Also, if i treat this as a derivative, then would...

I have an easy question which I've been thinking about for a while.. Lets say I want to take the derivative of a function y = f(x) with respect to x, we would get. dy/dx = f'(x). In the couple of books I've skimmed through, they all say that dy/dx is not a ratio but the notation that...

Dear All, I am unable to understand a simple mathematics relation. I spent 2-3 hours to Google multi-variable mathematics and have studied some concepts, still i am missing/confusing some basics. The problem I have at hand is following. Vector p can be written as p = (p1, p2, p3) = n(sin θ3...

Today I tried to show that rotational kinetic energy was equivalent to standard translational kinetic energy. So I started with kinetic energy, T = ∫dT. Then, because T=1/2mv^2, I substituted dT=1/2v^2dm and then because m=ρV, I substituted dm=ρdV. Then, after substituting v=ωr, I got the...

Hello! I've only just come across the Unruh Effect...so please bear with me! Say you have a long pole, and you spin the pole around its center. The ends of the pole would then be accelerating but the center of the pole wouldn't be. The Unruh Effect would seem to be saying the ends of the...

Take \(U(\eta) = u(x - ct)\) and the wave equation \(u_{tt} - u_{xx} = \sin(u)\). Then making the transformation, we have \[ (1 - c^2)U_{\eta\eta} = \sin(u). \] My question is the chain rule on the differential. \[ U_{\eta} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial\eta} +...

Hello, I currently have a problem with interpreting how this statement was interpreted: We have a rate of change which is dv/dt, and in the given notes, they transformed the expression dv/dt into dv/dx dx/dt (using the chain rule). Then, the whole expression simply turns into v dv/dx (as...

Consider the differential equation dx+ydy=0, the integration leads to (x2-x1)+(y2^2-y1^2)/2=0 (1) Suppose we know that y/x = const. Lest proceed to the following manipulation on the initial equation, by dividing by (x), then dx/x+(y/x)dy=0, now the integration gives...

Homework Statement http://i4.minus.com/jboxzSadIJVVoi.jpg Homework Equations Product rule; implicit differentiation. Volume of cylinder, V = pi(r^2)(h) The Attempt at a Solution dV/dt = 0 = pi[2r(dr/dt)(h) + (dh/dt)(r^2)] Solve the equation after plugging in r = 5; h = 8, and dh/dt =...

I just couldn't grasp the idea what are differentials,intuition behind them,applications of differentials? can anyone thoroughly tell me about it please? and why sometimes dy/dx is taken as a fraction?

Hi, I don't understand why in some texts they put that infinitesimal volume dV = dx dy dz. If V = x y z, infinitesimal volume should be dV = y z dx + x z dy + x y dz, by partial differentiation. Thanks

I am currently reading Calculus Made Easy by S. P. Thompson, and the author's idea of what it means for a variable to "vary" seems fundamentally different from my own, so I was hoping someone could help me correct my understanding. Here is the excerpt I'm having trouble with: Those...

Hello all, I have sort of a fundamental elementary calculus question. So, when trying to understand differentials, I always interpret is as change in the function corresponding to the change in the inputs. I always thought of these changes as "infinitesimal" changes and the differential for...

Homework Statement A welded railway train, of length 15km, is laid without expansion joints in a desert where the night and day temperatures differ by 50K. The cross sectional area of the rail is 3.6 x 10-3m2. A)What is the difference in the night and day tension in the rail if it kept at...

I'm taking a short Calculus session this summer and the teacher zooms through things. I still don't fully understand differentials. I know that derivative give you the slope of a function at any point. And I know that dy is a small change in y and dx is a small change in x and how they can be...

Hi all, In thermodynamics one often has equations like A dx + B dy = ∂f/∂x dx + ∂f/∂y dy From which follows A = ∂f/∂x B = ∂f/∂y Can anyone explain to me why this conclusion is necessary from a mathematical point of view, please? Here is my try: A dx + B dy = ∂f/∂x dx + ∂f/∂y...

From the two equations given below, find ∂s/∂V (holding h constant) and ∂h/∂V (holding r constant V = π*r^2*h, S = 2π*r*h + 2*π*r^2 Not entirely sure where to start...

I've been teaching myself some thermodynamics, and I've been thinking about solving the heat equation. \frac{\partial T}{\partial t} = K\frac{\partial ^2 T}{\partial x^2} I haven't taken a course in PDEs. I have noticed that if I assume an exponential solution, there are not non-decaying...

Homework Statement Here is the question along with the solution: Can anyone explain why the terms I circled in red are different? For the first term there is a negative sign but then the second term does not? Why did it disappear?

One side of a right triangle is known to be 20 cm long and the opposite angle is measured as 30°, with a possible error of ±1°. (a) Use differentials to estimate the error in computing the length of the hypotenuse. (Round your answer to two decimal places.) ±...cm(b) What is the percentage...

I'm currently teaching myself intermediate mechanics & am really struggling with the d'Alembert-based virtual differentials derivation for E-L. The whole notion of, and justification for, using 'pretend' differentials over a time interval of zero just isn't sinking in with me. And I notice...

I'm reading the second edition of John M. Lee's Introduction to Smooth Manifolds and he has a proposition that I'd like to understand better Let M, N, and P be smooth manifolds with or without boundary, let F:M→N and G:N→P be smooth maps and let p\inM Proposition: TpF : TpM → TF(p) is...

I recently did a problem with some electron constraint to move on a hoop. It kind of surprised me that you just could take the old Schrödinger-equation with and let your dx ->dβ, where β is the distance along the hoop. Saying it in a less mathematical way, isn't a differential distance along...

Homework Statement See thumbnail Homework Equations The Attempt at a Solution I'm not having trouble with the first part, just having trouble understanding why dQ is not exact but dS=dQ/T is. At first I was thinking that it had to do with the V in the dT part of the dQ...

Homework Statement Let R be a connected open region ( in the plane ). Suppose that F = (M,N) is a vector function defined on R and is such that for any ( piecewise smooth ) curve C in R : \int_C Fdp depends on only the endpoints of C ( that is, any two curves from P1 to P2 in R give...

Homework Statement I'm reviewing physics using Feynman's Lectures, and I'm finding that he frequently uses implicit differentiation in his lessons. This is unfortunate for me because I never got the hang of it beyond the simplest cases. I'm currently going through the proof that the...