Calculus Definition and 1000 Threads

  1. A

    Integrate [cosec(30°+x)-cosec(60°+x)] dx in terms of tan x

    I proceeded as follows $$\int\frac{2(\sqrt3-1)(cosx-sinx)}{2(\sqrt3+2sin2x)}dx$$ $$\int\frac{(cos(\pi/6)-sin(\pi/6))(cosx-sinx)}{(sin(\pi/3)+sin2x)}dx$$ $$\frac{1}{2}\int\frac{cos(\pi/6-x)-sin(\pi/6+x)}{sin(\pi/6+x)cos(\pi/6-x)}dx$$ $$\frac{1}{2}\int cosec(\pi/6+x)-sec(\pi/6-x)dx$$ [FONT=times...
  2. MAXIM LI

    Limit of probabilities of a large sample

    My first thought as well but I think the problem is deeper than that. I think that as the n tends towards infinity the probability of the the sample mean converging to the population mean is 1. Looking at proving this. By the Central Limit Theorem the sample mean distribution can be approximated...
  3. A

    How Does Basic Calculus Open Doors in Physics?

    I am quite interested in math and physics and keep on trying to discover new stuff. I recently learnt badic calculus and i find math cool
  4. Brix12

    I How do I format equations correctly? (Curl, etc.)

    A question in advance: How do I format equations correctly? Let's say $$\mathbf{k}\cdot\nabla\times(a\cdot\mathbf{w}\frac{\partial\,\mathbf{v}}{\partial\,z})$$ - a is a scalar Can I rewrite the expression such that...
  5. Safinaz

    How Does the Dirac Delta Function Identity Apply in Equation (27) Derivation?

    I need help to understand how equation (27) in this paper has been derived. The definition of P(k) (I discarded in the question ##\eta## or the integration with respect for it) is given by (26) and the definition of h(k) and G(k) are given by Eq. (25) and Eq. (24) respectively. In my...
  6. L

    Dirac delta function approximation

    Hi, I'm not sure if I have calculated task b correctly, and unfortunately I don't know what to do with task c? I solved task b as follows ##\displaystyle{\lim_{\epsilon \to 0}} \int_{- \infty}^{\infty} g^{\epsilon}(x) \phi(x)dx=\displaystyle{\lim_{\epsilon \to 0}} \int_{\infty}^{\epsilon}...
  7. A

    Electromagnetism: Force on a parabolic wire in uniform magnetic field

    I know the easier method/trick to solve this which doesn't require integration. Since parabola is symmetric about x-axis and direction of current flow is opposite, vertical components of force are cancelled and a net effective length of AB may be considered then ##F=2(4)(L_{AB})=32\hat i##...
  8. M

    How should I show that ## B ## is given by the solution of this?

    a) Consider the functional ## S[y]=\int_{0}^{v}(y'^2+y^2)dx, y(0)=1, y(v)=v, v>0 ##. By definition, the Euler-Lagrange equation is ## \frac{d}{dx}(\frac{\partial F}{\partial y'})-\frac{\partial F}{\partial y}=0, y(a)=A, y(b)=B ## for the functional ## S[y]=\int_{a}^{b}F(x, y, y')dx, y(a)=A...
  9. M

    How should I calculate the stationary value of ## S[y] ##?

    Consider the functional ## S[y]=\int_{1}^{2}x^2y^2dx ## stationary subject to the two constraints ## \int_{1}^{2}xydx=1 ## and ## \int_{1}^{2}x^2ydx=2 ##. Then the auxiliary functional is ## \overline{S}[y]=\int_{1}^{2}(x^2y^2+\lambda_{1}xy+\lambda_{2}x^2y)dx, y(1)=y(2)=0 ## where ## \lambda_{1}...
  10. Frabjous

    I Heavyside’s operational calculus vs. transforms

    Are there features of operational calculus (or operator methods) that are advantageous over transforms for DE? I know that the techniques are closely related.
  11. G

    Help me prove integral answer over infinitesimal interval

    In the book, I see the following: ##\int_{x_1}^{x_1 + \epsilon X_1} F(x, \hat y , \hat y') dx = \epsilon X_1 F(x, y, y')\Bigr|_{x_1} + O(\epsilon^2)##. My goal is to show why they are equal. Note that ##\hat y(x) = y(x) + \epsilon \eta(x)## and ##\hat y'(x) = y'(x) + \epsilon \eta'(x)## and...
  12. G

    Why Does Taylor's Theorem Use +O(ε) Instead of -O(ε)?

    I am trying to grasp how the last equation is derived. I understand everything, but the only thing problematic is why in the end, it's ##+O(\epsilon)## and not ##-O(\epsilon)##. It will be easier to directly attach the image, so please, see image attached.
  13. M

    How should I use the Jacobi equation to determine the nature of this?

    Here's my work: Let ## n>1 ## be a positive integer. Consider the functional ## S[y]=\int_{0}^{1}(y')^{n}e^{y}dx, y(0)=1, y(1)=A>1 ##. By definition, the Jacobi equation is ## \frac{d}{dx}(P(x)\frac{du}{dx})-Q(x)u=0, u(a)=0, u'(a)=1 ##, where ## P(x)=\frac{\partial^2 F}{\partial y'^2} ## and...
  14. P

    Problem involving space elevators

    (a) The length ##h = L## for which the tension is minimum is the length that corresponds to the geostationary orbit, where the angular velocity of the cable matches the angular velocity of the Earth. This is because at this point, the centrifugal force balances the gravitational force, and the...
  15. chwala

    Find the local maxima and minima for##f(x,y) = x^3-xy-x+xy^3-y^4##

    Ok i have, ##f_x= 3x^2-y-1+y^3## ##f_y = -x+3xy^2-4y^3## ##f_{xx} = 6x## ##f_{yy} = 6xy - 12y^2## ##f_{xy} = -1+3y^2## looks like one needs software to solve this? I can see the solutions from wolframalpha: local maxima to two decimal places as; ##(x,y) = (-0.67, 0.43)## ...but i am...
  16. chwala

    Find the dimensions that will minimize the surface area of a Rectangle

    My interest is on number 11. In my approach; ##v= xyz## ##1000=xyz## ##z= \dfrac{1000}{xy}## Surface area: ##f(x,y)= 2( xy+yz+xz)## ##f(x,y)= 2\left( xy+\dfrac{1000}{x} + \dfrac{1000}{y}\right)## ##f_{x} = 2y -\dfrac{2000}{x^2} = 0##...
  17. G

    I Integrating 1/x with units (logarithm)

    Hi. What exactly is happening mathematically when you integrate ##\frac{1}{x}## $$\int_a ^b \frac{1}{x} dx=\ln{b}-\ln{a}=\ln{\frac{b}{a}}$$ if there's units? Sure, they cancel if you write the result as ##\ln{\frac{b}{a}}##, but the intermediate step is not well-defined, so why should log rules...
  18. S

    Line Integral of circle in counterclockwise direction

    My attempt: Let ##x=a \cos \theta## and ##y=a \sin \theta## $$\int_{L} xy^2 dx-x^2ydy$$ $$=\int_{0}^{2\pi} \left( (a\cos \theta)(a\sin \theta)^2 (-a\sin \theta)-(a\cos \theta)^2 (a \sin \theta)(a\cos \theta)\right) d\theta$$ $$=-a^4 \int_{0}^{2\pi}\left( \sin^3 \theta \cos \theta+\cos^3 \theta...
  19. chwala

    Show that ##f(x,y)=u(x+cy)+v(x-cy)## is a solution of the given PDE

    Looks pretty straightforward, i approached it as follows, ##f_x = u(x+cy) + v(x-cy)## ##f_{xx}=u(x+cy) + v(x-cy)## ##f_y= cu(x+cy) -cv(x-cy)## ##f_{yy}=c^2u(x+cy)+c^2v(x-cy)## Therefore, ##f_{xx} -\dfrac{1}{c^2} f_{yy} = u(x+cy) + v(x-cy) - \dfrac{1}{c^2}⋅ c^2 \left[u(x+cy)+v(x-cy)...
  20. I

    Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates

    I want to prove that ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates where ##a,b,c \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch ) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1...
  21. I

    Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##

    I have to prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1 in Bloch's book) There exists a set...
  22. L

    Proving limits for roots and exponents

    Hi I have to prove the following three tasks I now wanted to prove three tasks with a direct proof, e.g. for task a)$$\sqrt[n]{n} = n^{\frac{1}{n}}= e^{ln(n^{\frac{1}{n}})}=e^{\frac{1}{n}ln(n)}$$ $$\displaystyle{\lim_{n \to \infty}} \sqrt[n]{n}= \displaystyle{\lim_{n \to \infty}}...
  23. gleem

    I Wish to Give Away a Textbook Set on The Calculus

    I have a two volume set of Differential and Integral Calculus textbooks by a well-known mathematician, his second edition (1959 printing) the last of twenty printings. This set was recommended by one of my math professors. Although it was my intention to read them it was not to be. Below is a...
  24. Tobi9242

    Mathematical model for drag on tether

    I have a model for airs density as a function of height I would imagine the speed can be describes as the angular velocity times length The coefficient of drag can be found online, seems to be around 1.17 for a cylinder It seems to me that im going to need an integral somewhere, but can't quite...
  25. E

    I have troubles finding the limit of this piecewise function

    I have troubles finding the limits at the designated points , should i only find the limit at infinity where f(x) has belongs to an interval containing inifinity? (sorry for english) and for the a this is what i attempted. i am unsure. Our textbook never talks about piecewise functions and...
  26. mcastillo356

    B Some questions regarding the integral of cos(ax), "a" not zero

    Hi PF, $$\int \cos ax\,dx,\quad a\in{\mathbb R-\{0\}}\quad x\in{\mathbb{R}}$$ Let's make $$u=ax,\quad du=adx$$ and apply $$\int \cos u\,du=\sin u+C$$ $$\frac{1}{a}\int \cos ax\,adx=\frac{1}{a}\sin u+C$$ Substituting the definition of u $$=\frac{1}{a}\sin ax+C$$ Doubts: (i) Have I written well...
  27. Z

    Calculation of moment of inertia of cylindrical surface

    Here is the homogenous paper rectangle And if we roll it we get a cylinder with base radius ##a##. It is not clear to me what "an axis through a diameter of the circular base means". Let's imagine such as axis is ##\alpha## in the following figure Then we have...
  28. Memo

    Help understanding this integral solution using trig substitution please

    Here's the answer: Could you explain the highlighted part for me? Thank you very much!
  29. Memo

    Integral involving powers of trig functions

    Could you check if my answer is correct? Thank you very much! Is therea simpler way to solve the math?
  30. I

    A Integration of trigonometric functions

    Was solving a problem in mathematics and came across the following integration. Unable to move further. Can somebody provide answer for the following ( a and b are constants ).
  31. chwala

    Find rate at which the liquid level is rising in the problem

    I was able to solve it using, ##\dfrac{dV}{dt} = \dfrac{dV}{dh}⋅\dfrac{dh}{dt}## With, ##r = \dfrac{h\sqrt{3}}{3}##, we shall have ##\dfrac{dV}{dh} = \dfrac{πh^2}{3}## Then, ##\dfrac{dh}{dt}= \dfrac{2×3 ×10^{-5}}{π×0.05^2}= 0.00764##m/s My question is can one use the ##\dfrac{dV}{dt} =...
  32. chwala

    Show that acceleration varies as cube of the distance given

    In my approach i have distance as ##(x)## and velocity as ##(x^{'})##, then, ##(x^{'}) = kx^2## where ##k## is a constant, then acceleration is given by, ##(x^{''}) = 2k(x) (x^{'})## ##(x^{''}) = 2k(x)(kx^2) ## ##(x^{''}) = 2k^2x^3##. Correct?
  33. Infrared

    Challenge Math Challenge Thread (October 2023)

    The Math challenge threads have returned! Rules: 1. You may use google to look for anything except the actual problems themselves (or very close relatives). 2. Do not cite theorems that trivialize the problem you're solving. 3. Do not solve problems that are way below your level. Some problems...
  34. S

    Existence of directional derivative

    My attempt: I have proved (i), it is continuous since ##\lim_{(x,y)\rightarrow (0,0)}=f(0,0)## I also have shown the partial derivative exists for (ii), where ##f_x=0## and ##f_y=0## I have a problem with the directional derivative. Taking u = <a, b> , I got: $$Du =\frac{\sqrt[3] y}{3 \sqrt[3]...
  35. S

    Find f(x,y) given partial derivative and initial condition

    My attempt: $$\frac{\partial f}{\partial x}=-\sin y + \frac{1}{1-xy}$$ $$\int \partial f=\int (-\sin y+\frac{1}{1-xy})\partial x$$ $$f=-x~\sin y-\frac{1}{y} \ln |1-xy|+c$$ Using ##f(0, y)=2 \sin y + y^3##: $$c=2 \sin y + y^3$$ So: $$f(x,y)=-x~\sin y-\frac{1}{y} \ln |1-xy|+2 \sin y + y^3$$ Is...
  36. M

    Determine whether ## S[y] ## has a maximum or a minimum

    a) The Euler-Lagrange equation is of the form ## \frac{d}{dx}(\frac{\partial F}{\partial y'})-\frac{\partial F}{\partial y}=0, y(a)=A, y(b)=B ##. Let ## F(x, y, y')=(y'^2+w^2y^2+2y(a \sin(wx)+b \sinh(wx))) ##. Then ## \frac{\partial F}{\partial y'}=2y' ## and ## \frac{\partial F}{\partial...
  37. F

    Insights Epsilontic – Limits and Continuity

    Continue reading...
  38. mcastillo356

    B Integral of cosecant function: understanding different approaches

    Hi, PF Trigonometric Integrals "The method of substitution is often useful for evaluating trigonometric integrals" (Calculus, R. Adams and Christopher Essex, 7th ed) Integral of cosecant...
  39. P

    How to get general solution via Green's function?

    I'll start with a characterization of the Green's function as a fundamental solution to a differential operator. This theorem is given in Ordinary Differential Equations by Andersson and Böiers. ##E(t,\tau)## is known as the fundamental solution to the differential operator ##L(t,D)##, also...
  40. S

    Multivariable calculus problem involving partial derivatives along a surface

    I just wanted to know if my solution to part (b) is correct. Here's what I did: I took the partial derivative with respect to x and y, which gave me respectively. Then I computed the partial derivatives at (-3,4) which gave me 3/125 for partial derivative wrt x and -4/125 for partial derivative...
  41. J

    Introduction to Modern Astrophysics -- What are the prerequisites?

    I was recently recommended this book and told it was a standard textbook at an upper undergraduate level or lower graduate level. Well that's certainly above my level, but specifically what would be the prerequisites? I've no formal math training but self taught calculus at a level somewhere...
  42. S

    Unit tangent vector and curvature with arc length parameterization

    (a) $$\frac{ds}{dt}=|r'(t)|$$ $$=\sqrt{(x(t))^2+(y(t))^2+(z(t))^2}$$ $$=\frac{2}{9}+\frac{7}{6}t^4$$ $$s=\int_0^t |r'(a)|da=\frac{2}{9}t+\frac{7}{30}t^5$$ Then I think I need to rearrange the equation so ##t## is the subject, but how? Thanks Edit: wait, I realize my mistake. Let me redo
  43. M

    Calculus ## G(y, z)=z\frac{\partial}{\partial z}F(y, z)-F(y, z) ##?

    a) Observe that ## \frac{\partial}{\partial z}F(y, z)=y^{n-1}\cdot \frac{2z}{2\sqrt{y^2+z^2}}=\frac{zy^{n-1}}{\sqrt{y^2+z^2}} ##. This means ## G(y, z)=\frac{z^2\cdot y^{n-1}}{\sqrt{y^2+z^2}}-y^{n-1}\cdot \sqrt{y^2+z^2}=\frac{z^2\cdot...
  44. M

    Proofs about the second-order linear differential equation?

    Proof: (i) Consider the second-order linear differential equation ## \frac{d^2u}{dx^2}+\frac{fu}{2}=0, f=f(x) ##. Then ## u''+\frac{f}{2}u=0\implies r^2+\frac{f}{2}=0 ##, so ## r=\pm \sqrt{\frac{f}{2}}i ##. This implies ## u_{1}=c_{1}cos(\sqrt{\frac{f}{2}}x) ## and ##...
  45. A

    A Summing over continuum and uncountable numerocities

    Here I want to address of the question if it is possible to make a sum over an uncontable set and discuss integration rules involving uncountably infinite constants. I will provide introduction in very condensed form to get quicker to the essense. Conservative part First of all, let us...
  46. mcastillo356

    B Trigonometric substitution, a case I'd like to share

    Hi, PF First I will quote it; next the doubts and my attempt: "In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expresions. In calculus, trigonometric substitution is a technique for evaluating integrals. (...) [FONT=times new roman]Case I...
  47. P

    I Lipschitz continuity of vector-valued function

    I'm reading Ordinary Differential Equations by Andersson and Böiers, although this is more related to multivariable calculus. There is a Lemma regarding Lipschitz continuity which I have a question about. Below ##\pmb{f}:\mathbf{R}^{n+1}\to \mathbf{R}^n ## is a vector-valued function defined by...
  48. KungPeng Zhou

    A question about definite integrals and series limits

    In my opinion , if it can be shown that this is a monotonically bounded sequence, one can confirm that there is a limit. First,we know $$ \frac{1-x^{4n}}{1+x^{2}}dx=(1-x^{2}) (1+x^{2}) ^{n-1}=(1-x^{4}) ^{n-1}(1+x^{2}).$$ According to the integral median theorem,we can get $$a_n=(2- \sqrt{3} )...
  49. KungPeng Zhou

    How Do You Solve Calculus Homework Problem Six Using LaTeX?

    How to solve the sixth problem?
  50. bhobba

    Insights Beginner's Guide to Precalculus, Calculus and Infinitesimals

    [url="https://www.physicsforums.com/insights/precalculus-calculus-and-infinitesimals/"]Continue reading...
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