Derivative Definition and 1000 Threads

Homework Statement I have somewhat general question about time derivative of a vector. If we have r=at2+b3 it's easy to find instantaneous acceleration and velocity(derivative with respect to dt) v=2at+3bt2 a=2a+6bt But consider this position vector r=b(at-t2) where b is constant vector and a...

I've been thinking about something recently: The notation d2x/d2y actually represents something as long as x and y are both functions of some third variable, say u. Then you can take the second derivatives of both with respect to u and evaluate d2x/du2 × 1/(d2y/du2). Now I think it's also...

In the equation regarding an array of masses connected by springs in wikipedia the step from $$\frac {u(x+2h,t)-2u(x+h,t)+u(x,t)} { h^2}$$ To $$\frac {\partial ^2 u(x,t)}{\partial x^2}$$ By making ##h \to 0## is making me wonder how is it rigorously demonstrated. I mean: $$\frac {\partial ^2...

Homework Statement Hey guys, Consider the U(1) transformations \psi'=e^{i\alpha\gamma^{5}}\psi and \bar{\psi}'=\bar{\psi}e^{i\alpha\gamma^{5}} of the Lagrangian \mathcal{L}=\bar{\psi}(i\partial_{\mu}\gamma^{\mu}-m)\psi. I am meant to find the expression for \partial_{\mu}J^{\mu}. Homework...

Recently I started with multivariable calculus; where I have seen concepts like multivariable function, partial derivative, and so on. A week ago we saw the following concept: directional derivative. Ok, I know the math behind this as well as the way to compute the directional derivative through...

Homework Statement (Self study.) Several sources give the following for the Riemann Curvature Tensor: The above is from Wikipedia. My question is what is \nabla_{[u,v]} ? Homework Equations [A,B] as general purpose commutator: AB-BA (where A & B are, possibly, non-commutative operators)...

Homework Statement r = 2\cos(\theta) Homework EquationsThe Attempt at a Solution Hello, please do not evaluate. Why do textbook state that the derivative of the polar function (symbolic) is dy/dx and not dr/d\theta? It is a function of theta, then why is the derivative dy/dx? Idea: Even...

Hi, My first challenge was not very popular so I bring you another one. Let us define $$f(x)=\dfrac{sin(x)}{x}$$ for $$x>0$$. Prove that for every $$n\in \mathbb{N}$$, $$|f^{(n)}(x)|<\dfrac{1}{n+1}$$ where $$f^{n}(x)$$ denotes the n-th derivative of $$f$$

Homework Statement Let A be a set of critical points of the function f(x). Let B be a set of roots of the equation f''(x)=0. Let C be a set of points where f''(x) does not exist. It follows that B∪C=D is a set of potential inflection points of f(x). Q 1: Can there exist any inflection points...

I am new to this forum, i don't know if it's here i should post this simple question. I have to find the peak of the function: ##\frac{x}{\sqrt{x^2+R^2}(x^2+R^2)}=\frac{x}{(x^2+R^2)^{3/2}}## I differentiate: ##\left( \frac{x}{(x^2+R^2)^{3/2}} \right)'=\frac{(x^2+R^2)^{3/2}+x\left(...

When applying the least action I see that a term is considered total derivative. Two points are not clear to me. We say that first $$\int \partial_\mu (\frac {\partial L}{\partial(\partial_\mu \phi)}\delta \phi) d^4x= \int d(\frac {\partial L}{\partial(\partial_\mu \phi)}\delta \phi)= (\frac...

Hey guys, This is really confusing me cos its allowing me to create factors of 2 from nowhere! Basically, the first term in the Lagrangian for a real Klein-Gordon theory is \frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi). Now let's say I wana differentiate this by applying the...

Hi guys, So I've ended up in a situation where I have \partial_{\mu}\Box\phi. where the box is defined as \partial^{\mu}\partial_{\mu}. I'm just wondering, is this 0 by any chance...? Thanks!

b1. Homework Statement Let ##c## and ##z## denote complex numbers. Then 1. When a branch is chosen for ##z^c##, then ##z^c## is analytic in the domain determined by that branch. 2. ##\frac{d}{dz} z^c = c z^{c-1}## Homework EquationsThe Attempt at a Solution In regards to number one, we have...

I'm working on a ODE with initial conditions y(2)=4 and y'(2)=1/3. I solved it to be y=\frac{c_1}{|x-6|^8} + c_2|x-6|^{\frac{2}{3}}. How do I apply the second initial condition? I'm stuck at taking the derivative.

Im doing a question on functionals and I have to use the Euler lagrange equation for a single function with a second derivative. My problem is I don't know how to evaluate \frac{d^2}{dx^2}(\frac{\partial F}{\partial y''}). Here y is a function of x, so y'=\frac{dy}{dx}. I know this is probably...

A ladder 10 ft long rests against a vertical wall. Let be the angle between the top of the ladder and the wall and let be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to $\theta$ when $\theta...

Homework Statement Derivative question f=f(x) and x=x(t) then in one book I find \frac{d}{dx}\frac{df}{dt}=\frac{d}{dx}(\frac{df}{dx}\frac{dx}{dt}) =\frac{dx}{dt} \frac{d^2 f}{dx^2} Homework EquationsThe Attempt at a Solution Not sure why this is correct? \frac{dx}{dt} can depend of f for...

Homework Statement If w = w(x, y, z) is given implicitly by F(x, y, z, w) = 0, find a formula for both ∂w/∂z and ∂^2w/∂y∂z . You may assume that each function is sufficiently differentiable and anything you divide by during the process of your solution is non-zero. The Attempt at a Solution I...

Homework Statement 1) I am having trouble with the questions, "Use the logarithmic derivative to find y' when y=((e^-x)cos^2x)/((x^2)+x+1) Homework Equations (dy/dx)(e^x) = e^x (dy/dx)ln(e^-x) = -x ? The Attempt at a Solution First I believe I put ln on each set of terms (Though I don't know...

Homework Statement Find the partial derivative of a*cos(xy)-y*sin(xy) with respect to y. Homework Equations None. The Attempt at a Solution The answer is -ax*sin(xy)-sin(xy)-xy*cos(xy). I know that I need to treat x as constant since I need to take the partial derivative with respect to y...

Homework Statement The revenue function for a product is r = 8x where r is in dollars and x is the number of units sold. the demand function is q = -1/4p + 10000 where q units can be sold when selling price is p. what is dr/dp? Homework Equations r=pq The Attempt at a Solution I substituted...

My https://www.amazon.com/dp/0073532320/?tag=pfamazon01-20 (p. 176 Example 7.1) pointed out that an investment ##p(t) = 100\,2^t## (##t## in year) that doubles the capital every year starting with an initial capital of $100, has an (instantaneous) rate-of-change ##\frac{\text{d}}{\text{d}t} p(t)...

Hi at 1 Hour and 9 minutes this professor makes a derivation which i do not understand He is lecturing on Newtonian mechanics and states that if dv/dt = a (acceleration) Then v*dv/dt = a*v And then he says that this is the same as d(v^2/2)/dt But I just can't undrestand how he did...

Homework Statement Show U^a \nabla_a U^b = 0 Homework Equations U^a refers to 4-velocity so U^0 =\gamma and U^{1 - 3} = \gamma v^{1 - 3} The Attempt at a Solution I get as far as this: U^a \nabla_a U^b = U^a ( \partial_a U^b + \Gamma^b_{c a} U^c) And I think that the...

\mathcal{L}_M(g_{kn}) = -\frac{1}{4\mu{0}}g_{kj} g_{nl} F^{kn} F^{jl} \\ \frac{\partial{\mathcal{L}_M}}{\partial{g_{kn}}}=-\frac{1}{4\mu_0}F^{pq}F^{jl} \frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})=+\frac{1}{4\mu_0} F^{pq} F^{lj} 2 \delta^k_p \delta^n_j g_{ql} I need to know how...

Homework Statement Show that if F is an antiderivative of f on [a,b] and c is in (a,b), then f cannot have a jump or removable discontinuity at c. Hint: assume that it does and show that either F'(c) does not exist or F'(c) does not equal f(c). 2. The attempt at a solution I attempted a proof...

I'm trying to come up with an expression for \partial y / \partial x where z = f(x,y). By observation (i.e. evaluating several sample functions), the following appears to be true: \begin{equation*} \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial...

What would be some important properties of a universe where Force = Mass * Jerk and objects stay in constant acceleration until acted upon by a net force? (if we ignore the fact that objects would reach the speed of light, and just deal with classical mechanics)

Homework Statement So this is a problem that I am at a complete loss with. The question asked is, give an equation, using the quotient or power rule that derivative is equal to either sec(x) or cot(x). It doesn't matter which one, sec(x) or cot(x), just as long as the initial equation's...

Hello, we haven't really covered partial differentiation in my maths course yet, but it has come up a few times in mechanics where the 'grad' operator is being introduced, so I'm trying to learn about it myself. I'm looking at the partial derivatives section in "Mathematical Methods In The...

Homework Statement If f(x) = x^5*cos(x^6) find f40(0) and f41(0) The Attempt at a Solution So we are supposed to get the Taylor series and use that to get the value of the derivatives I just manipulated the Taylor series for cosx to get the one for this. Would the value be the coefficient?

Hi, I'm trying to find the eigenvalues and eigenvectors of the operator ##\hat{O}=\frac{d^2}{d\phi^2}## Where ##\phi## is the angular coordinate in polar coordinates. Since we are dealing with polar coordinates, we also have the condition (on the eigenfunctions) that ##f(\phi)=f(\phi+2\pi)##...

Homework Statement We are given that ##f(z) = u(x,y) + iv(x,y)## and that the function is differentiable at the point ##z_0 = x_0 + iy_0##. We are asked to determine the directional derivative of ##f## 1. along the line ##x=x_0##, and 2. along the line ##y=y_0##. in terms of ##u## and...

Homework Statement Show that f'(x) = k/x Homework Equations f is defined from zero to infinity f(xy) = f(x) + f(y) f'(1) = k f(1) = 0 f(x+h) = f(x) + f (1+h/x)The Attempt at a Solution [/B] I know i can write f'(x) = f'(1)/x but that's all I've got so far...

What does it mean when I have to find the second derivative of a circle at a given point? (Implicit diffing) In specifics, the equation is 9x2 +y2 =9 At the point (0,3) You don't really need the rest at all, but it was just my process. This seems to make no sense. first D'v 18x+2yy'=0 Second...

Hi, So I'm working through a bunch of problems involving gradient vectors and derivatives to try to better understand it all, and one specific thing is giving me trouble. I have a general function that defines a change in Temperature with respect to position (x,y). So for example, dT/dt would...

Have a function $$f(x)=4x^3-x^4$$ Found the x values are X -1, 0, 1, 2, 3 , 4, f(Y) -5, 0, 3, 16, 27, 0 i Need to find $$f^{\prime}(x)$$ and find where it incteases and decreases??$$f`(x)= 3*4x^2-4x^3=4x^2(3-x)$$ what to Next?

Homework Statement A police car is parked 50 feet away from a wall. The police car siren spins at 30 revolutions per minute. What is the velocity the light moves through the wall when the beam forms angles of: a) α= 30°, b) α=60°, and c) α=70°? This is the diagram...

Do you guys know a place where I can find a proof of the formula \frac{d^{(n)}f(z)}{dz^{n}} = \frac{n!}{2\pi i}\oint \frac{f(z)dz}{(z- z_{0})^{n+1}} Thanks

Homework Statement Find \frac{\partial f}{\partial x} if f(x,y)=\cos(\frac{x}{y}) and y=sinx Homework Equations See above The Attempt at a Solution For \frac{\partial f}{\partial x} I calculated -\frac{1}{y}\sin(\frac{x}{y}) which comes out as \frac{-\sin(\frac{x}{\sin(x)})}{sinx} and this...

Homework Statement W(t) = W(y1, y2) find the Wronskian. Equation for both y1 and y2: 81y'' + 90y' - 11y = 0 y1(0) = 1 y1'(0) = 0 Calculated y1: (1/12)e^(-11/9 t) + (11/12)e^(1/9 t) y2(0) = 0 y2'(0) = 1 Calculated y2: (-3/4)e^(-11/9 t) + (3/4)e^(1/9 t)Homework Equations W(y1, y2) = |y1 y2...

The function f: R → R is: f(x) = (tan x) / (1 + ³√x) ; for x ≥ 0, sin x ; for (-π/2) ≤ x < 0, x + (π/2) ; for x < -π/2 _ For the interval (0,∞), we are interested in f such that f(x) = (tan x) / (1 + ³√x) ; for x ≥ 0 f(x) = tan x / (1 + x¹ʹ³)            (1 + x¹ʹ³)•sec²x −...

Steps for finding the derivative of 4/sqrt{x}

##dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy## I'm confused as to how the total derivative represents the total change in a function. My own interpretation, which I know is incorrect, is that ##\frac{\partial z}{\partial x} dx## represents change in the x...

Homework Statement Suppose a can, after an initial kick, moves up along a smooth hill of ice. Make a statement concerning its acceleration. A) It will travel at constant velocity with zero acceleration. B) It will have a constant acceleration up the hill, but a different constant acceleration...

I was trying to see what is the covariant derivative of a covector. I started with $$ \nabla_\mu (U_\nu V^\nu) = \partial_\mu (U_\nu V^\nu) = (\partial_\mu U\nu) V^\nu + U_\nu (\partial_\mu V^\nu) $$ since the covariant derivative of a scalar is the partial derivative of the latter. Then I...

Is the derivative of a function a differential equation? I guess it would be because it involves a derivative, right? Would the solution to the equation just be the original function? Is solving a differential equation just another way of integrating? Like with finding solutions of separable...

Let's say I have Fourier series of some function, f(t), f(t)=\frac{a0}{2}+\sum_{n=1}^{\infty}(an\cos{\frac{2n\pi t}{b-a}}+bn\sin{\frac{2n\pi t}{b-a}}), where a and b are lower and upper boundary of function, a0=\frac{2}{b-a}\int_{a}^{b}f(t)dt, an=\frac{2}{b-a}\int_{a}^{b}f(t)cos\frac{2n\pi...

Homework Statement Given the functions Q(v,w) and R(v,w) [/B] K = v(dQ/dv)r and L = v(dQ/dv)w Show that (1/v)K = (1/v)L + (dQ/dw)v (dW/dv)r I have the problem attached if for clarity of the information. Homework Equations I assume everything is given in the problem. The Attempt at...