1. C

    Differential Integration Problem

    Attempt at solution: Writing the chain rule for ## E(V,T) ##: ## dE = \frac{\partial E}{\partial T}dT + \frac{\partial E}{\partial V}dV ## Then, integrating the differential: ## \int{ dE } = \int{ \frac{\partial E}{\partial T}dT } + \int{ \frac{\partial E}{\partial V}dV } ## If I put the...
  2. Protea Grandiceps

    I X variable in damping force equation for damped oscillation?

    Hi, for ease of reference this posting is segmented into : 1. Background 2. Focus 3. Question 1. Background: Regarding (one, linear, second-order, homogeneous, ordinary, differential) equation describing the force in a non-driven, damped oscillation: F = m.a = -k.x - b.v F =...
  3. Q

    I Deriving the spherical volume element

    I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using $$dxdydz = \left (\frac{\partial x}{\partial r}dr +...
  4. Cathr

    I Integrating a differential

    Let's say we have ##df=2xy^3dx + 3x^2y^2dy## - this is an exact differential. In integrating, to find f, can we write ## f = \int 2xy^3 \, dx + \int 3x^2y^2 \, dy = 2x^2y^3 + C ## Or am I getting it wrong?
  5. L

    B Line Integral, Dot Product Confusion

    From my interpretation of this problem (image attached), the force applied to the point charge is equal and opposite to the repulsive Coulomb force that that point charge is experiencing due to the presence of the other point charge so that the point charge may be moved at a constant velocity. I...
  6. M

    A Differentiability of a function between manifolds

    Hello, let $$M^n \subset \mathbb{R}^N$$ $$N^k \subset \mathbb{R}^K$$ be two submanifolds. We say a function $$f : M \rightarrow N$$ is differentiable if and only if for every map $$(U,\varphi)$$ of M the transformation $$f \circ \varphi^{-1}: \varphi(U) \subset \mathbb{R}^N \rightarrow...
  7. C

    I Christoffel symbols knowing Line Element (check my result)

    Hi! I'm asked to find all the non-zero Christoffel symbols given the following line element: ds^2=2x^2dx^2+y^4dy^2+2xy^2dxdy The result I have obtained is that the only non-zero component of the Christoffel symbols is: \Gamma^x_{xx}=\frac{1}{x} Is this correct? MY PROCEDURE HAS BEEN: the...
  8. komarxian

    Differential Equations: Solve the following

    1. Homework Statement Solve the following differential equations/initial value problems: (cosx) y' + (sinx) y = sin2x 2. Homework Equations I've been attempting to use the trig ID sin2x = 2sinxcosx. I am also trying to solve this problem by using p(x)/P(x) and Q(x) 3. The Attempt at a...
  9. komarxian

    Differential Equations, solve the following: y^(4) - y'' - 2y' +2y = 0

    1. Homework Statement Solve the following differential equations/initial value problem: y^(4) - y'' - 2y' +2y = 0 Hint: e^-x sinx is a solution 2. Homework Equations I was attempting to solve this problem by using a characteristic equation. 3. The Attempt at a Solution y'''' -y'' -2y' +...
  10. M

    I Heat equation plus a constant

    I have seen how to solve the heat equation: $$ \frac{ \partial^2 u(x,t) }{\partial x^2} = k^2 \frac{ \partial u(x,t) }{\partial t} $$ With boundary conditions. I use separation variables to find the result, but i dont know how to solve the equation plus a...
  11. S

    How to calculate total cross section from differential cross section

    I was doing the calculations for this: http://fermi.la.asu.edu/PHY531/cylinder/index.html But I can't figure out how to go from \frac{d\sigma}{d\phi} to the total cross section. My guess was that you did the integral from \phi=0 to \phi=2\pi, but that's not helping since I can't tell either how...
  12. H

    Find the derivative of this function

    1. Homework Statement $$y = x.\log_e {\sqrt{x}}$$ 2. Homework Equations f(x) = g(x) h(x) f ' (x) = g ' (x) . h (x) - h ' (x) . g(x) 3. The Attempt at a Solution $$y = x .\log_e {\sqrt {x}}$$ $$y '(x) = 1.ln \sqrt{x} + \frac{1}{2} $$ the right answer is $$ y ' = \log_{10} {\sqrt{x}} +...
  13. Adgorn

    I Differentials of order 2 or bigger that are equal to 0

    So I've seen in several lectures and explanations the idea that when you have an equation containing a relation between certain expressions ##x## and ##y##, if the expression ##x## approaches 0 (and ##y## is scaled down accordingly) then any power of that expression bigger than 2 (##x^n## where...
  14. Alexander350

    B Solving a differential equation with a unit vector in it

    I need to solve: \dot{\mathbf{r}}=-kv\hat{r} - \dot{\mathbf{r}_s} However, I do not know how to deal with the fact that there is a unit vector. How can this be done? \dot{\mathbf{r}_s} is a constant vector.
  15. Alexander350

    B Modelling water tanks

    I tried using the Bernoulli equation to solve this. I took two points at the surface of the water in both the containers and formed this equation: gh_{b}=\frac{1}{2}v^2+gh This is assuming that the velocity of the water in the large tank is approximately zero and using the fact that both the...
  16. A

    Help with mass-spring modeling problem

    1. Homework Statement Suppose at time zero, the bob was drawn upward four units from the equilibrium position, let C=2, K=2, m=1 lbm, initial speed=2 unit/sec find an expression for body's position. and in the solution it says: y''+6y'+5y=0 my question is: from where does the numbers (6) and...
  17. W

    Differential Equations: LRC

    1. Homework Statement How does one show that q(t) is indeed a solution? 2. Homework Equations 3. The Attempt at a Solution My current idea is that i should come up with any form of solution, like q = Acos(ωt), and slot it in the RHS. Reason being that if q is indeed a solution, the result...
  18. S

    B Some help understanding integrals and calculus in general

    So in differential calculus we have the concept of the derivative and I can see why someone would want a derivative (to get rates of change). In integral calculus, there's the idea of a definite integral, which is defined as the area under the curve. Why would Newton or anyone be looking at the...
  19. O

    How to solve the differential equation for driven series RLC circuit?

    1. Homework Statement It is the driven series RLC circuit. It is given in the following images. It is from the section 12.3 in this note. 2. Homework Equations The differential equation, as given by 12.3.3, is ##L \frac{d^2 Q}{d t^2} + R \frac{d Q}{d t} + \frac{Q}{C} = V_0 \sin{(\omega...
  20. I

    Vector differential displacement - Magnetic vector potential

    1. Homework Statement http://imgur.com/a/k7fwG Find the vector magnetic potential at point P1. 2. Homework Equations Vector magnetic potential given by: $$ d \bar{A} = \frac{\mu I d\bar{l'}}{4 \pi | \bar{r} - \bar{r'} | } $$ 3. The Attempt at a Solution I split up the problem in 3 parts...
  21. M

    Properties of Solutions of Matrix ODEs

    1. Homework Statement We assume from ODE theory that given a smooth A: I → gl(n;R) there exists a unique smooth solution F : I → gl(n;R), defined on the same interval I on which A is defined, of the initial value problem F' = FA and F(t0) = F0 ∈ gl(n;R) given. (i) Show that two solutions Fi...
  22. M

    Frenet Frame

    1. Homework Statement The Frenet frame of a curve in R 3 . For a regular plane curve (and more generally for a regular curve on a 2-dimensional surface - e.g. the 2-sphere above) we could construct a unique adapted frame F. This is not the case for curves in higher dimensional spaces. Besides...
  23. M

    Various Properties of Space Curve...*Really need help*

    1. Homework Statement Let γ : I → R3 be an arclength parametrized curve whose image lies in the 2-sphere S2 , i.e. ||γ(t)||2 = 1 for all t ∈ I. Consider the “moving basis” {T, γ × T, γ} where T = γ'. (i) Writing the moving basis as a 3 × 3 matrix F := (T, γ × T, γ) (where we think of T and...
  24. J

    Find A and B so that F(x) is a Differentiable Function

    1. Homework Statement Find the values of a and b that make f a differentiable function. Note: F(x) is a piecewise function f(x): Ax^2 - Bx, X ≤ 1 Alnx + B, X > 1 2. Homework Equations 3. The Attempt at a Solution Made the two equations equal each other. Ax^2 - Bx = Alnx + B Inserting x=1...
  25. M

    Proving basic linear ODE results

    1. Homework Statement Please bear with the length of this post, I'm taking it one step at a time starting with i) Let A: I → gl(n, R) be a smooth function where I ⊂ R is an interval and gl(n, R) denotes the vector space of all n × n matrices. (i) If F : I → gl(n, R) satisfies the matrix ODE...
  26. Math Henry

    Differentiation question

    1. Homework Statement Summarizing: two civilizations hate each other, one of the civilizations throws a curse at the second. The second civilization succumbs to chaos and has a change in Population each week of ΔP= -√P. That is: Pn = Pn-1-√Pn-1 2. Homework Equations Considering that the...
  27. Duke Le

    Where is wrong in this proof for rotational inertia ?

    1. Homework Statement Prove the formula for inertia of a ring (2D circle) about its central axis. 2. Homework Equations I = MR^2 Where: M: total mass of the ring R: radius of the ring 3. The Attempt at a Solution - So I need to prove the formula above. - First, I divide the ring...
  28. W

    2nd Order Linear ODE-Derivation of system-issue

    1. Homework Statement How exactly they combined equation1 and equation2 and got that system? I don't get that part. 2. Homework Equations A*(dy/dt)= -k*y eq1 A*(dz/dt)=ky-kz eq2 3. The Attempt at a Solution I tried substituting the 1st ky in the 2nd equation and then differentiating but...
  29. J

    Analysis Books on solving DE with infinite series?

    Hi folks, I was wondering if there are books that explain how to solve differential equations using infinite series. I know it is possible to do it since Poincaré used that method. Do you know which ones are the best? I find books on infinite series but they talk just about series...
  30. J

    Matrix riccati differential equation using matlab

    1. Homework Statement 2. Homework Equations 3. The Attempt at a Solution