Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread uses in science, engineering, and economics.In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus (plural calculi) is a Latin word, meaning originally "small pebble" (this meaning is kept in medicine – see Calculus (medicine)). Because such pebbles were used for counting (or measuring) a distance travelled by transportation devices in use in ancient Rome, the meaning of the word has evolved and today usually means a method of computation. It is therefore used for naming specific methods of calculation and related theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus.
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
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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}##...
Homework Statement
The problem (Spivak's Calculus, chapter 6, problem 17):
"Let ##f## be a function with the property that every point of discontinuity is a removable discontinuity. This means that ##\underset {y \rightarrow x} {\lim} {f(y)}## exists for all ##x##, but ##f## may be...
Homework Statement
I have to find length of the curve: y2 = (x-1)3 from (1,0) to (2,1)
Homework Equations
s = ∫ √(1 + (f '(x) )2 ) dx where we have integral from a to b
The Attempt at a Solution
I'm bit confused:
I'm thinking of writing function regarding x, f(x)...
Homework Statement
Hi everyone, I'm currently making my way through Spivak's calculus and got stuck in question 41 of chapter 5. It's important to note that at this point, the book has only reached the subject of limits (haven't reached continuous functions, derivatives, integrals, series...
I've been researching for the calculus behind geodesic domes, and specifically calculus related to parametric surfaces. I've found http://teachers.yale.edu/curriculum/viewer/new_haven_06.04.05_u#f, but unfortunately, it comes short of providing me the most needed information, and so far I...
Homework Statement
I'm currently working through Spivak independently and have reached the problems at the end of ch. 1.
The problem is:
Prove that if 0 < a < b , then a < \sqrt{ab} < \frac{a+b}{2} < b
Homework Equations
Spivak's properties P1 - P12
The Attempt at a Solution
I was...
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?
Hi there all,
I'm currently taking a course in Multivariable Calculus at my University and would appreciate any recommendations for a textbook to supplement the lectures with. Thus far the relevant material we've covered in a Single Variable course at around the level of Spivak and some Linear...
I tried my best but the result doesn't match the effort I've put in.
First when I know my result I freaked out and questioned my competent.
Am I mathematically-inept?
Am I cut out for a Physics Phd?
End up with ~70% on this course and I'm taking Calc 2 this December.
I need to...
If I have a limit of a series then how can I convert it into integral. I know to convert a sum into an integral there must be Δx multiplied to each term and this must go zero. Can you please explain me the conversion of limit of series (normal series with no Δx) into an integral.
Thank you.
I am currently a junior in high school and recently, my guidance counselor has been asking me a lot of questions about what I want to major in. Within the last 6-8 months, I have been leaning heavily towards a math major. That was until I started my calc class this year...
I'm in AP Calc BC...
I'm struggling with the math used in my college's calc-based honors physics class, even though I've taken calculus 1.
---What are some good books/resources to learn the math used in introductory physics?---
Preferably, it'd be nice if the math was taught using examples in physics. Having that...
I don't understand a good portion of the non-algebraic math behind much of the physics in my first semester college class. I understand everything with algebra, and can solve all problems, but I don't understand the relationships with vector cross/dot products, calculus derivations, DE, etc...
I've been working on this one for a little bit, and I know I really just need to use the chain rule to solve it, but I can't seem to figure out how to set it up properly. Probably a dumb question, but I could really use some help on this!
Homework Statement
Hello,
a bowl is created when rotating the function
f(x) =
\begin{cases}
0, & 0 \leq x \lt 6 \\
(12/\pi)arcsin(x-6), & 6\leq x \leq 7
\end{cases}
around the y-axis. Find the height (h) and the volume (V) of the bowl.Homework EquationsThe Attempt at a Solution
So, I graphed...
Homework Statement
Hello, I need some feedback on whether this reasons is correct.
consider the series
Examine the series for absolute convergence.
Homework EquationsThe Attempt at a Solution
How I have solved this, using the limit comparison test:
we have:
introducing
we have that...
Homework Statement
Hello,
I´ve added an image where the task is shown. I am wondering about the integral bounds here. When doing the substitution, shouldn't the bounds be inf and 0? When using that u = ln x, i.e. u(0) =inf, u(1) = 0.
Homework EquationsThe Attempt at a Solution
Homework Statement
Find the sum of the series
Homework EquationsThe Attempt at a Solution
Not sure exactly where to start. If I move 3 outside the sum I'm left with 3*sigma(1/n*4^n), which I can rewrite to 3*sigma((1/n)*(1/4)^n), which party looks like a geometric series..Any tips?
Homework Statement
Hello,
I need to find an expression for the sum of the given power series
The Attempt at a Solution
I think that one has to use a known Maclaurin series, for example the series of e^x. I know that I can rewrite
, which makes the expression even more similar to the...
Afternoon, I have been working on this problem for awhile now but have been stuck on a certain point, and once I set everything equal to each other I end up with the same thing for example: x=x, y=y, & z=z
Find the minimum and maximum values of the function f(x,y,z)=3x+2y+4z subject to the...
Hi, I want to start learning physics as I have never studied it in my life but have now taken an interest towards it. I have quite literally zero knowledge on physics, so assume I haven’t taken middle school or high school courses on physics, as I haven’t. I’m 17 currently, but that shouldn’t...
I have been stumped on this problem and I am probably overthinking it as usual. The problem deals with the Lagrange Multipliers:
Use Lagrange multipliers to find the point (a,b) on the graph of y=e^(9x), where the value (a,b) is as small as possible.
I have found the gradient for both when...
Homework Statement
Find the points on the surface xy^2z^3=2 that are closest to the origin
Homework EquationsThe Attempt at a Solution
x,y,z=/= 0, as when x,y,z = 0 it is untrue. Right?? Otherwise, I am very unsure as to how to approach this problem. Should I be taking partial derivatives to...
Homework Statement
The question is to solve the inexact equation by turning it into exact.the equation is ##( x + y + 4 ) d x + ( - x + y + 6 ) d y = 0##
Where "x" and "y" are variable.
2. Homework Equations [/B]
1.(x+y+4)=m and (-x+y+6)=n
2.Integrating Factor =##\frac { 1 } { x ^ { 2 } + y...
Homework Statement
Suppose light is prepared in a coherent superposition of linear horizontal polarization and linear vertical polarization. What is the resulting polarization according to Jones calculus if it passes through:
a linear polarizer at a 45-degree angle (0 degrees would be...
Homework Statement
Let D be the triangle with vertices (0,0), (1,0) and (0,1). Evaluate:
∫∫exp((y-x)/(y+x))dxdy for D
by making the substitutions u=y-x and v=y+x
Homework EquationsThe Attempt at a Solution
So first I found an equation for y and x respectively:
y=(u+v)/2 and x=(v-u)/2
Then...
The question goes like: find the SA of the portion S of the cone z^2 =x^2 +y^2 where z>=0 contained within the cylinder y^2+z^2<=49
this is my attempt using the formula for SA, I could switch to parametric eqns, but even then I'd have hard time setting up limits of integration.
Homework Statement
Given the graph of f(x) shown below, find the value of the integral.
Photo attached.
Homework Equations
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∫23 5x·f(x2)dx
The Attempt at a Solution
[/B]
I tried integration by parts to simplify the problem, but finding the integral of the composite function (f(x2))...
I have the first and second orders that I use in a magnetic simulator, but i need the thirth also to do also with magnetic cylinders accordingly paper:
Do anybody have it in any code? I should pass to C++
Homework Statement
The equation 4x = (1/3)*cos(3x) has a solution on the interval [0,1]. Find an approximative solution by replacing the right hand side with a Taylor polynomial of degree 2 around 0.
Homework EquationsThe Attempt at a Solution
So as I understand the task we should find a...