Calculus Definition and 1000 Threads

Hello everybody. If anyone could help me solve the calculus problem posted below, I would be greatful. Task: Evaluate the moment of inertia with respect to Oz axis of the homogeneous solid A Bounded by area - A: (x^2+y^2+z^2)^2<=zSo far I was able to expand A: [...] so that I receive...

Problem Statement: A known mass at a know velocity collides on a spring of known stiffness. What is the equation that governs the deceleration of the mass, so that the force on the spring could be found? Relevant Equations: 1/2 m*V^2 = 1/2*k*x^2 + 1/2*m*(Vo)^2 Kinetic energy of mass before...

Summary: Mechanics problem related with Calculus (differential equations) Hi everyone, I would like some help in that task, if anyone would be willing to help :) Namely I have a problem from particle dynamics. "D:" means given info... so, D: m,g,h,b, miu. We're looking for v0 and S as given...

I want a book that isn't too wordy, explains things concisely, and covers single-variable and multi-variable calculus. I regret that I forgot so much of the stuff I learned in Calculus III and IV (especially the important theorems I learned but forgot in the latter), and now I want to review it...

I am attempting to solve an ODE using a Calculus add-in for Excel. I am an industry professional and I have not even thought about Differential Equations in 8 years. The equation that I am attempting to solve is in the form: (1) The ODE solver that I am using solves equations of the form...

Let: ##\displaystyle f=\int_{V'} \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'## where ##V'## is a finite volume in space ##\mathbf{r}=(x,y,z)## are coordinates of all space ##\mathbf{r'}=(x',y',z')## are coordinates of ##V'## ##|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}##...

Hello, I have been given a general equation that is dS/dt = (n-x-rS)(L-1) and it needs to be rearranged in order to for the subject to be t. I have spent a while trying to find a way to attempt this but with no luck so I will leave this here. Thanking You, BSLAHi

I actually don't know how to proceed. I tried something like this The left side of the equation equals to $$\delta(\int_a^b F(x)dx)=\delta f(x) |_{a}^{b}$$ where ##f'(x)=F(x)## However $$\delta f(x) |_{a}^{b}=f'(x)\delta x dx|_{a}^{b} = \delta (F(b)-F(a))$$ where ##f'(x)=F(x)##. For the...

For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.

My volume integral is... $$\pi\int y^2 dx$$ My surface area integral is... $$2\pi\int y \sqrt {1+x'^2} dy$$I'm fairly sure the variable of integration on my volume and surface area integrals has to be the same, is that right? But when I change the variable in the surface area integral to...

Hello all. I've come across some math which consists of just applying the basic ideas of calculus (derivatives and integrals) onto discrete functions. (The link: http://homepages.math.uic.edu/~kauffman/DCalc.pdf ) The discrete derivative with respect to n is defined as ## \Delta_n f(n) = f(n+1)...

I have attached a word document demonstrating the working out cos i was too lazy to learn how Latex primer works and writing it like I did above would've been too hard too read. I tried to make it as understandable as possible, presenting fractions as ' a ' instead of ' a / b ' . ------ b

Hello, I need help with question #2 c) from the following link (already LateX-formatted so I save some time...): https://wiki.math.ntnu.no/_media/tma4135/2017h/tma4135_exo1_us29ngb.pdf I do understand that the a0 for both expressions must be the same, but what about an and bn? I don't...

Why are the red circled Del operators not combining to become 'Del-squared' to cancel out the second term to give a net result of 0?

Quotient rule: z= f/g ------ z'= (f'g - g'f)/g^2 starting with finding the derivative in respect to x, i treated y^2 as constant 'a': z'= [(a*2x*cos a*x^2)(sin a*x^2) - (- a*2x*sin ax^2)]/cos(a*x^2)^2= [(a*2x*cos a*x^2)(sin a*x^2)+(a*2x*sin ax^2)]/cos(a*x^2)^2 For the derivative in respect to y...

<Moderator's note: Member has been warned to show some effort before an answer can be given.> Hello all. This is my first post in this forum, I am asking for your understanding. I have a problem with the calculus task and I stuck in a dead endso I managed to find a solution on the internet. I...

I split this to get \begin{equation} \int ^{\infty} _{0} \dfrac{e^{ax}}{(1+e^{ax})(1+e^{bx})} \ dx - \int ^{\infty} _{0} \dfrac{e^{bx}}{(1+e^{ax})(1+e^{bx})} \ dx \end{equation} Then I tried to solve the first term (both term are similars). The problem is that I made a substitution (many ones...

Hello, I am working through Spivak for self study and sharpening my math skills. I have become stuck on an exercise. What I need to show is the following: $$ (a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = \sum_{j = 0}^{n + 1} \binom{n+1}{j} a^{n-j + 1}b^{j} $$ My attempt, starting from...

Two weeks ago I had no idea on how to code using Python. Now I have completed an online course on functions, loops and strings. However, in that course I did not practice using the specific library called Sympy. Besides, I will use Python in the Physics-Math background, for solving problems like...

Homework Statement Homework Equations The Attempt at a Solution [/B] The solution to this problem is known. I want to use this exercise as a model to understand how to proceed when calculating the surface area of a geometric figure. Question: 1) Why do we differentiate with...

Homework Statement Homework Equations $$F = \nabla \phi$$ The Attempt at a Solution Let's focus on determining why this vector field is conservative. The answer is the following: [/B] I get everything till it starts playing with the constant of integration once the straightforward...

Homework Statement Homework Equations $$F = \nabla \phi$$ $$| F | = \sqrt{F \cdot F}$$The Attempt at a Solution I want to compute ##| F | = \sqrt{F \cdot F}## $$| F | = \sqrt{F \cdot F} = \sqrt{\frac{(km)^2 (r-r_0)^2}{|r-r_0|^6}} = \sqrt{\frac{(km)^2}{(r-r_0)}} =...

Homework Statement Question: If ##r_+ \neq r_{++}## and ## g(r=r_+) \neq g(r=r_{++}) ## When is it fulfilled that ## d g (r=r_+) \neq d g (r=r_+) ## ? Homework Equations ##r_+ \neq r_{++}## ## g(r=r_+) \neq g(r=r_{++}) ## The Attempt at a Solution I tried computing ## dg(r_+) =...

Homework Statement Hello, i am trying to do find the Fourier series of abs(sin(x)), but have some problems. As the function is even, bn = 0. I have calculated a0, and I am now working on calculating an. However, when looking at the solution manual, they have set up one calculation for n > 1...

I am looking for a book for learning Python so as to compute matrices, eigenvalues, eigenvectors, divergence, curl (i.e vector calculus). If you also have online recommendations please feel free to write them.

Homework Statement Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n## Evaluate: ##f^{(8)}(4)## Homework Equations The Taylor Series Equation The Attempt at a Solution Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the...

I have a strange question. It's strange because I don't need a correct answer. I need an answer that seems correct and leads to predictable results. I'm making a multiplayer computer game where the players fire cannons in outer space. The cannon shells will move through the gravitational fields...

I need to make an integral to fine the speed of the earth. Say the radius is 6378137 meters. How would I account for things closer to the axis that have a radius of 0.0001 meters? I don't think I can just take the speed at the radius. So I found that the Earth rotates at 6.963448857E-4 revs/min...

In Calculus II, we're learning about solids of revolutions and computing their volumes. I'm unsure when to apply the appropriate methods and how to make the correct partitions. Please tell me if my reasoning is correct: The disk/washer method is applied when your partitions are perpendicular...

So I found the linear velocity by using the circumference of the Earth which I found to be 2pi(637800= 40014155.89meters. Then the time of one full rotation was 1436.97 minutes, which I then converted to 86164.2 seconds. giving me the linear velocity to be 465.0905584 meters/second. I know that...

In Calculus II, we're currently learning how to find the area of a bounded region using integration. My professor wants us to solve a problem where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (c,d) with nothing else given. I...

Hi, I'm in 12 grade, and I always was a decent student but recently I became more fascinated by learning and I want to learn some math and physics beyond high school level. One or two weeks ago I started the calculus course on mit ocw (18.01sc) (at session 17 right now, right before lecture 7)...

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 +...

1)An airplane heads due east at 300 mph through a tailwind whose velocity is given by $w=\langle20,20\rangle$ How fast is the tailwind blowing? In what direction? How fast is the plane flying? In what direction?Answer: The tailwind is blowing at 28.2843 mph approx.in $45^{\circ}$ direction...

Homework Statement Let x, y and z satisfy the state function ##f(x, y, z) = 0## and let ##w## be a function of only two of these variables. Show the following identities: $$\left(\frac{\partial x}{\partial y}\right )_w \left(\frac{\partial y}{\partial z}\right )_w =\left(\frac{\partial...

Homework Statement A charge q1 is at rest at the origin, and a charge q2 moves with speed βc in the x-direction, along the line z = b. For what angle θ shown in the figure will the horizontal component of the force on q1 be maximum? What is θ in the β ≈ 1 and β ≈ 0 limits? (see image) Homework...

I need to learn about Hamiltonian mechanics involving functional and functional derivative... Also, I need to learn about generalized real and imaginary Hamiltonian... I only learned the basics of Hamiltonian mechanics during undergrad and now those papers I read show very generalized version...

Alright so I was just messing around with Lagrangian equation, I just learned about it, and I had gotten to this equation of motion: Mg*sin{α} - 1.5m*x(double dot)=0 I am trying to get velocity, and my first thought was to integrate with dt, but I didn't know how to. And I'm not even sure it's...

I saw the roadmap of my university of and there is no calculus for biophysics, as soon as there is no calculus for medicine, as for example there is in the US. My question is if there is calculus for biophysics in Europe and in the US. Adendum: there is only calculus in Brazil for medicine in...

Homework Statement The acceleration of a particle given a=A√t where A=2.0 m/s5/2. At t=0, v=7.5 m/s and x=0. (a) What is the speed as a function of time? (b) What is the displacement as a function of time? (c) What are the acceleration, speed, and displacement at t=5.0s. Homework EquationsThe...

Homework Statement A disk with a 0.4 m radius starts from rest and is given an angular acceleration α = (10θ2/3)rad/s2 , where θ is in radians. Determine the magnitude of the normal (centripetal and tangential components of a point P on the rim of the disk when t = 4s. Homework Equations α =...

Homework Statement I have an empty 2-Liter bottle. It contains 3 g of air inside with an initial air pressure of 1105 mb. When I open it (which is an adiabatic process), I release the pressure which is instantaneous. The pressure then becomes standard atmospheric pressure. What is the...

Homework Statement [/B] I am trying to determien the characteristic function of the function: $$ f(x)= ae^{-ax}$$ $$\therefore E(e^{itx}) =\int_0^\infty e^{itx}ae^{-ax} dx = a \cdot \frac{e}{it-a} |_0 ^ \infty $$ But I am not sure how to evaluate the integral. Wolfram alpha suggests this...

Homework Statement I am stuck on number 11 on my homework. Homework Equations Not Sure The Attempt at a Solution I know this has to have something to do with Riemann's Sum, but I am lost on where to start. I started by putting numbers in for p, but I think that is wrong.

Homework Statement Hi, everyone. I came across a basic calculus problem concerning movement of 2 problems, I've attempted to solve it using vector analysis and got 1 answer, and then solved it with differentiation and got a different answer. I'll show a version of it I made which is a bit more...

<Moderator's note: Moved from a technical forum and thus no template.> $$\lim_{x \to 0} \cos(\pi/2\cos(x))/x^2$$ I tried to evaluate the limit this way, $$\lim_{x \to 0} \cos(\pi/2\cdot1)/x^2$$ since $$\cos0=1$$ $$\lim_{x \to 0} \cos(\pi/2\cdot1)/x^2=\lim_{x \to 0} 0/x^2$$ Now apply...

Hello everyone, Going through calculus study, there is a vague point regarding polynomials I'd like to make clear. Say there's a polynomial ##f## with a root at ##a## with multiplicity ##2##, i.e. ##f(x)=(x-a)^2g(x)## where ##g## is some other polynomial. I define ##h(x)=\frac {f(x)} {x-a}##...

When have a function and I know by investigation of that it getting "bigger and bigger" or getting "smaller and smaller", how could I know that in infinity it continue by that way always?

I have an analytical function F of the discrete variables ni, which are natural numbers. I also know that the sum of all ni is constant and equal to N. N also appears explicitly in F, but F is not a function of N. F exists in a coordinate system given by the ni only. Should I carry out the...

Homework Statement An electrostatic field ## \mathbf{E}## in a particular region is expressed in cylindrical coordinates ## ( r, \theta, z)## as $$ \mathbf{E} = \frac{\sin{\theta}}{r^{2}} \mathbf{e}_{r} - \frac{\cos{\theta}}{r^{2}} \mathbf{e}_{\theta} $$ Where ##\mathbf{e}_{r}##...