Differentials Definition and 236 Threads

Homework Statement About how accurately must the interior diameter of a 10-m high cylindrical storage tank be measured to calculate the tank's volume to within 1% of its true value? Homework Equations V=\frac{5}{2}\pi l^{2}, where V is volume and l is diameter. dV=5\pi l \ dl The...

Using z = y/x to transform the given homogeneous differential equation into a differential equation in z and x. By first solving the transformed equation, find the general solution of the original equation, giving y in terms of x. z = \frac{y}{x} \rightarrow y = xz \rightarrow \frac{dy}{dx} =...

solve dy/dx = x(1-x) I got y = (x^2)/2 - (x^3)/3 + C however in the solutions they've gotten: where did t come from?

It's been awhile since I've taken a differential equations course, so I just could not wrap my head around this one. Homework Statement I was given a lot of variables but it boils down to a partial differential equation that looks like: pT/pt = A*p^2T/px^2 + B*f(x) I am not looking for...

. Homework Statement The tension T in the string of the yo-yo is given by: T=(mgR)/(2r^2+R^2) where m is the mass of the yo-yo and g is the acceleration due to gravity. Use differentials to estimate the change in tension if R is increased from 3cm to 3.1cm and r is increased from 0.7cm...

I am looking at the derivation of the capstan friction equation and there is a term in there which the derivation claims can be neglected; my question is: why can it be neglected? dT*sin(dθ/2) source: http://www.jrre.org/att_frict.pdf

Homework Statement Hi Say I have the equality \frac{df(x)}{dx} = \frac{dg(z)}{dz} where f and g are two functions that are well-behaved such that I can take their derivate. The variables x and z are both real, and run from -∞ to ∞. In this case, am I allowed to divide out the differentials...

What is the change of variables using differentials trick K&K are referring to here...

Homework Statement The proof begins: Suppose that F is conservative. Then a scalar field ε(r) can be defined as the line integral of F from the origin to the point r. So ∫F dot dr = ε(r), where the limits of integration are from 0 to r. The next step, however, eludes me: From the...

How important are differentials and linear approximation in the study of calculus? I mean the dy=f'(x)dx stuff. It seems simple but I always thought you couldn't treat the dy/dx as a fraction? And can the integration...formulas be derived without using differentials (think it's the...

I am trying to reproduce results of a paper. The model is: dS = (v-y-\lambda_1)Sdt + \sigma_1Sdz_1 \\ dy = (-\kappa y - \lambda_2)dt + \sigma_2 dz_2 \\ dv = a((\bar{v}-v)-\lambda_3)dt + \sigma_3 dz_3 \\ dz_1dz_2 = \rho_{12}dt \\ dz_1dz_3 = \rho_{13}dt \\ dz_2dz_3 = \rho_{23}dt \\...

Homework Statement Find a differential of second order of a function u=f(x,y) with continuous partial derivatives up to third order at least.Hint: Take a look at du as a function of the variables x, y, dx, dy: du= F(x,y,dx,dy)=u_xdx +u_ydy. Homework Equations The Attempt at a Solution I'll be...

Homework Statement Here is the problem with the solution:http://dl.dropbox.com/u/64325990/MATH%20253/Capture.PNG I don't understand how dV is the error. Isn't the error the actual value - the estimated value? In other words, ΔV-dV?

I'm brushing up on differentiating multi-variable functions subject to a constraint and was curious about the notation. In particular, why the derivatives change from complete to partial derivates. I've illustrated the question with an example, below. My specific question w.r.t. the example is...

I have a simple question about differentials. I have been taught two ways to find the differential and my questions is in what situations do I use each one? simply speaking these are the 2 ways 1.) just take the partials of each component function and throw them in a matrix 2.) Let f be the...

Hi, This is a hard question to ask, because it's so vague . . . I have real trouble getting my head around using differentials to derive equations. Stuff like the fundamental differential equation of hydrostatics, eulers conservation of momentum equation in fluid mechanics, and bernoulli's...

Any books discussing the formula of d^2Z and d^3Z? Are they liked that? Anyone saw them before? Z(x, y)\\\\dZ=Z_xdx+Z_ydy\\d^2Z=Z_{xx}(dx)^2+2Z_{xy}dxdy+Z_{yy}(dy)^2+Z_xd^2x+Z_yd^2y\\d^3Z=Z_{xxx}(dx)^3+3Z{xxy}(dx...

Homework Statement Suppose that y(x) is the solution to the initial problem, y'=y(1-x), y(1)=e find y(2)Homework Equations The Attempt at a Solution This is my initial attempt: \frac{dy}{dx}=y(1-x) \frac{dy}{y}=(1-x)dx i then integrated both side to get: lny=-ln(1-x)+C and here's the problem...

Alright, I know my algebra, geometry, and plane trigonometry. I've been calculating derivatives for a while now, but I wish to get to a higher level of differentiation. Say, differential equations. Could somebody explain how to solve a differential equation?

I have quite some trouble thinking why we are allowed to manipulate differentials as we see fit when solving differential equations. I usually think of the derivative as the fundamental object upon which differentials are based. With this in mind I wince when I see derivatives appear separately...

Homework Statement Given that \mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V find a function G such that \mathrm{d}G = V \mathrm{d} p - S \mathrm{d} t . I'm not sure where to start - how are the two related? Could someone please give me a clue of how to start this off? 3. Attempt at the...

EDIT: I just found this thread, which handily has the same exact problem. The OP says that dh is = 0 though, and I don't quite understand why he does that. Homework Statement Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height 12 cm if the tin is...

I cannot figure out how to do this problem completely: If U =x3y, find \frac{dU}{dt} if x5 + y = t and x2 + y3 = t2. I know that I am using the chain rule here and I have the partial derivates of U: \frac{∂U}{∂x} = 3x2y \frac{∂U}{∂y} = x3 So far I have the equation given below...

Folks, I am trying to see how the torque is split 50:50 in a standard axle differential mathematically or via a free body diagram. Does anyone know? Thanks

Folks, Is it true to say for standard "open" differentials that when a car is turning a corner that the torque is equal on both wheels but the power will be different based on the expression P=TW, assuming that there is sufficient traction for all wheels, ie no slip? thanks

Homework Statement If you have a point at x = c and a function f(x), then I know Δy = f(c + Δx) - f(c). Also, dy = f'(c)dx. However, I am uncertain of the origin of dy = f'(x)dx. I want to say: f(c + Δx) - f(c) = f'(c)(x-c) was simplified to dy = f'(c)dx where f(c + Δx) - f(c) = dy...

Homework Statement 1 1/2 ∫ ∫ e-x2 dx dy 0 y/2 Homework Equations ***Graph equation*** The Attempt at a Solution I graphed the function and they made a horizontal strip. However, I can't seem to find the right function with a vertical strip, which is where I'm stuck.

I asked a question a few weeks ago about 'splitting' the derivative. The thread can be found https://www.physicsforums.com/showthread.php?p=3581188#post3581188" The answer to why it can not be split is because dx does not exist, it is simple a notation and not a fraction. However, I just...

Hi. I recently started studying differential equations, so bear with me. I started out with the following equation: y'' - 10y' -61y = xe^{-x} I know the method for solving these, but the thing I don't understand perhaps isn't the differential eq. part, rather the simplification of the...

Homework Statement Four positive numbers, each less than 40, are rounded to the first decimal place and then multiplied together. Use differentials to estimate the maximum possible error in the computed product that might result from the rounding. Homework Equations...

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...

Homework Statement Use differentials to estimate the amount of tin in a closed tin can with diameter 8cm and height 12cm if the tin is 0.04cm thick. Homework Equations If: z=f(x,y) then dz = f_{x}(x,y)dx+f_{y}(x,y)dy The Attempt at a Solution Perhaps my problem here...

What is the difference? I was pretty bored last night so I got onto Yahoo Answers and answered a few calculus questions. It was a simple integration by parts question: \intxsin(x) dx I solved as: u = x du = dx dv = sin(x) dx v = -cos(x) uv - \intvdu -xcos(x) + \intcos(x)dx =...

Homework Statement Four positive numbers, each less than 30, are rounded to the first decimal place and then multiplied together. Use differentials to estimate the maximum possible error in the computed product that might result from the rounding. So our function of four variables would be ...

Homework Statement Find the general solution for the following systems of equations ( 2 0 ) ( 0 2 ) Homework Equations (A-Ix) c1e^at[]+c2e^bt[] The Attempt at a Solution when attempting to find the eigenvalues I come up with (2) so plugging back into get the vectors you come up...

Homework Statement Just have to ask one more question :redface:. I have two problems which I don't understand in Kleppner Kolenkow. They seem to contradict each other in my view. The first is 3.9 and the second is 3.10. 3.9 A freight car of mass M contains a mass of sand m. At t = 0 a...

Homework Statement dv/dt = (k+v2)/h Homework Equations k is a constant, and v and h are variables where h is independent of v but v is dependent of h (v is a function of h and t). The Attempt at a Solution dv/(k+v^2) = dt/h. The problem I have is with dealing with the right side...

Hey guys, We skimmed a chapter on differential eqns in my semester 2 calculus class and we have a worksheet to fill out. I'm having trouble setting up the last 3 problems in this sheet. http://dl.dropbox.com/u/85600/DIFF_HW.PDF" I'm pretty sure It's all just algebraic manipulation to...

Homework Statement Four positive numbers, each less than 50, are rounded to the first decimal place and then multiplied together. use differentials to estimate the maximum possible error in the computed product that might result from the rounding. The Attempt at a Solution i need an...

Homework Statement Problem 7.29: http://cas.umkc.edu/physics/wrobel/phy240/Homework%20%205.pdf Homework Equations dr= R d(theta) The Attempt at a Solution I don't understand how to get R d(theta) = dr from the last part of the question, any explanation about how it works is...

Homework Statement a = (y^3 + y)dx + (xy^2 + x)dy = A1dx + A2dy Characterise the set of points in R2 which can be joined to (1,1) by a null curve. Homework Equations If v = z(t) is a piecewise continuous curve, and dv/dt lies in the null space of [A1(x(t)),A2(x(t))] then it is a null curve...

So my calculus isn't as sharp as I'd like it, and I am having trouble solving this diff eq, and I know the correct method is double integration. I also know the general solution to be T(r) = A*ln(r) + B , A and B are constants Thanks in advance

Hey everyone! This has always been a source of confusion for me that everyone seems to play around with dx and dy as if they were variables while in many sources it was stated that they are purely symbolic. For example, in the integral, dx is purely symbolic. If I'm not misunderstanding, dx is...

A hypothetical substance has a compressibility k = a /V and a volume expansivity B = 2bT /V , where a and b are constants and V is the molar volume. Show that the equation of state is: V = bT2 - aP + constant To be honest I'm not entirely sure what I'm actually supposed to be doing with...

Homework Statement Prove that if z=z(x,y) is invertible that: (dz/dx)(dy/dz)(dx/dy)=-1 where the d's represent partial differentiation not total differentiation Homework Equations The Attempt at a Solution I guess you start with the 6 total derivatives and substitute them...

Hi, my question is sort of a general work problem. using that work is equal to the integral from x initial to x final of F dot dl, I'm having trouble trying to visualize why this works for a spring. assuming, for example, a spring is stretched from equilibrium, the force of the spring is...

Hi everyone, This question is a bit involved but it pertains to calculating the differential of a variable substitution used in the proof of the convolution theorem (http://en.wikipedia.org/wiki/Convolution_theorem) Consider \int f(t) \int g(s - t) ds dt. If we use the substitution...

I just want to make sure I understand differentials in the context of thermal physics. One of the big statements of thermodynamics is the conservation of energy in terms of the state variables U,T,S,V,P: dU = T dS - P dV. What does this really MEAN though? Is there any way to...

I am reading a math review in my thermodynamics text and I a little confused by this. Here is the excerpt: I am confused by the part where it says If they selected x = (y, z) then isn't that saying that x is dependent on y? So how can we just turn around and say y = y(x, z) ? That...

Area of right triangle with hypotenuse H is A=(1/4)H^2sin(2theta) where theta is one of the acute angles. Use differentials to estimate the relative errors of the area A if H=4cm and theta is measured to be 30 degrees with an error of measurement of 15 minutes of arc. note: a minute...