In finance, a derivative is a contract that derives its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be used for a number of purposes, including insuring against price movements (hedging), increasing exposure to price movements for speculation, or getting access to otherwise hard-to-trade assets or markets.
Some of the more common derivatives include forwards, futures, options, swaps, and variations of these such as synthetic collateralized debt obligations and credit default swaps. Most derivatives are traded over-the-counter (off-exchange) or on an exchange such as the Chicago Mercantile Exchange, while most insurance contracts have developed into a separate industry. In the United States, after the financial crisis of 2007–2009, there has been increased pressure to move derivatives to trade on exchanges.
Derivatives are one of the three main categories of financial instruments, the other two being equity (i.e., stocks or shares) and debt (i.e., bonds and mortgages). The oldest example of a derivative in history, attested to by Aristotle, is thought to be a contract transaction of olives, entered into by ancient Greek philosopher Thales, who made a profit in the exchange. Bucket shops, outlawed in 1936, are a more recent historical example.
This week, I've been assigned a problem about a 3-sphere. I am confused how to approach this problem and any comments would be greatly appreciated.
(a) - would I be correct to assume the metric G is simply the dot product of two vector fields with dx^2 dy^2 du^2 and dv^2 next to their...
Is a connection the same thing as a covariant derivative in differential geometry?
What Is the difference between a covariant derivative and a regular derivative?
If you wanted to explain these concepts to a layperson, what would you tell them?
Let $p(x)$ be a polynomial with real coefficients. Prove that if $p(x)-p'(x)-p''(x)+p'''(x)\ge 0$ for every real $x$, then $p(x)\ge 0$ for every real $x$.
I've been able to find the tangent line with most equations, but I don't have any idea how to do it with a range of values instead of being given a singular value.
I can't help but feeling these days that I don't actually understand where most of the maths I use comes from. Unfortunately, I can't remember whether this is due to the fact that I didn't take my studies seriously until the end of undergrad, or rather that these things were never actually...
Hello again. I am sorry I got another problem when learning QFT regarding the Lorentz transformation of derivatives.
In David Tong's notes, he says
I am not sure how to transform the partial derivatives. Explicitly, should ##\frac {\partial} {\partial x ^{\mu}}## transform to ##\frac...
Does anyone know any good research on this topic? I'm basically looking for information on what would be solving integral and differential equations in which the unknown you need to solve for is the level of a integral or derivative in the equation. For example F'1/2(u)+F'x(u)=F'1/3(u) where the...
Definition: Let f be a differentiable real-valued function on ##\mathbf{R}^3##, and let ##\mathbf{v}_P## be a tangent vector to it. Then the following number is the derivative of a function w.r.t. the tangent vector
$$\mathbf{v}_p[\mathit{f}]=\frac{d}{dt} \big( \mathit{f}(\mathbf{P}+ t...
I research about coordinate systems and I found the following informations about transformation.
Now, if I replace arctan (x/y) (according to the picture above) to φ, I think I can solve. But if I can do this, then what will be replaced to ψ? I mean, I know just taking partial derative about...
Mentor note: Fixed the LaTeX in the following
I have the following statement:
\begin{cases} u=x \cos \theta - y\sin \theta \\ v=x\sin \theta + y\cos \theta \end{cases}
I wan't to calculate:
$$\dfrac{\partial^2}{\partial x^2}$$
My solution for ##\dfrac{\partial^2}{\partial...
I have a vector field which is originallly written as $$ \mathbf A = \frac{\mu_0~n~I~r}{2} ~\hat \phi$$ and I translated it like this $$\mathbf A = 0 ~\hat{r},~~ \frac{\mu_0 ~n~I~r}{2} ~\hat{\phi} , ~~0 ~\hat{\theta}$$(##r## is the distance from origin, ##\phi## is azimuthal angle and ##\theta##...
Here is an image of the structure
I know that cyclopropyl methyl carbocation is exceptionally stable due to an effect so called dancing resonance which takes place because of lot of strain in cyclopropyl ring and vacant p orbital of Carbon attached with the ring.
So I think this is a similar...
In defining the Wirtinger (aka Cauchy-Riemann) linear operators, often used in signal analysis and in proofs of complex derivatives and the Cauchy-Riemann equations, one assumes differentiability in the real sense. This assumption is usually seen as obvious in the complex analysis setting...
I'm in a first-year grad course on statistical mechanics and something about multivariable functions that has confused me since undergrad keeps popping up, mostly in the context of thermodynamics. Any insight would be much appreciated!
This is a general question, but as an example imagine...
So in my lecture notes on Differential Equations, it states that a first order ODE is exact if A(x,y)dx + B(x,y)dy = 0 and ∂A/∂y = ∂B/∂x. Okay I accept this definition.
Then, there is a sentence like this:
Our goal is to find the function V(x,y) satisfying
Adx + Bdy = dV = ∂V/∂x(dx) +...
##\frac {\delta I[f]} {\delta f(x_o)} = \int_a ^b \delta(x-x_o) \, dx## with a=-1 and b=+1
## -1 \leq x_o \leq +1 ## vs ## -1 \lt x_o \lt +1 ##, 0 otherwise. Which is correct and does it matter when doing integration by parts?
Hi,
I just need some (hopefully) quick calculus help.
I have the following:
##(y\frac {\partial } {\partial z}(z\frac{\partial f} {\partial x}))##
After it is expanded this is the solution:
##(yz\frac {\partial^2 f} {\partial z \partial x} + y\frac{\partial f} {\partial x} \frac{\partial z}...
I have seen the usage of a term which corresponds to changing energy per time and that would be power. Are there any such corresponding terms that can be described as 2nd time rate of change of energy, a sort of acceleration of energy. Is such a notion non existent in physics and completely...
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...
I need some help in formulating a proof of Proposition 8.12 ...
Proposition 8.12 reads...
I am wondering why the two methods below give different answers. If I multiply z through the second bracket I get $$(\frac{d}{dx} +x)(-\frac{dz}{dx} + xz)$$which, on expansion, yields $$-\frac{d}{dx}\frac{dz}{dx} -x\frac{dz}{dx} + \frac{d(xz)}{dx} + x^{2}z = -\frac{d^{2}z}{dx^{2}} + x^{2}z +...
You basically just take the second derivative of the given function and multiply it by the original then multiple everything by m. I just don’t understand how the second derivative would be negative.
ok just posted an image due to macros in the overleaf doc
this of course looks like a sin or cos wave and flips back and forth by taking derivatives
looks like a period of 12 and an amplitude of 3 so...
but to start I was not able to duplicate this on desmos
altho I think by observation alone...
Well, I just had this thought earlier, and I want to share it. Here it is.
So, we all know about inertia, right? The resistance to acceleration, or change in motion. Well, there is also a concept about derivatives of acceleration, mainly jerk and yank. If you don't know, jerk is said to be the...
In Lagrangians we often take derivatives (##\frac{\partial}{\partial (\partial_{\mu}\phi)}##) of terms like ##(\partial_{\nu}\phi \partial^{\nu}\phi)##. We lower the ##\partial^{\nu}## term with the metric and do the usual product rule. My question is why do we do this? Isn't...
I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ...
I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ...
I need help with some aspects of Examples 1.1 and 1.2, Section 1.2, Chapter III ...
Examples 1.1 and 1.2, Section 1.2...
I can show that ##\frac{d}{dt} \langle \psi (t) \vert X^2 \vert \psi (t) \rangle = \frac{1}{m} \langle \psi (t) \vert PX+XP \vert \psi (t) \rangle##.
Taking another derivative with respect to time of this, I get ##\frac{d^2}{dt^2} \langle \psi (t) \vert X^2 \vert \psi (t) \rangle = \frac{i}{m...
As I have said before, I am in calculus class for the first time. I am doing really well in the class, however because of how my mind works, I’m always asking questions to know more, even when it’s too advanced for me. I just like to ponder and think about “what if” I know it’s probably not good...
I was reading about differentiable manifolds on wikipedia, and in the definition it never specifies that the differentiable manifold has a metric on it. I understand that you can set up limits of functions in topological spaces without a metric being defined, but my understanding of derivatives...
Now, I understand how to use the quotient rule for derivatives and everything. I do not struggle with using it, my question is mostly about the formula itself...I very much enjoy WHY we do things in math, not just “here’s the formula, do it”...Here is the formula for the quotient rule of...
Homework Statement: The homework problem is included below, but I am looking at the derivatives of vectors.
Homework Equations: I have the properties of derivatives below, but not sure they help me here...
Two frames measure the position of a particle as a function of time: S in terms of x and t and S', moving at constant speed v, in terms of x' and t'. The acceleration as measured in frame S is $$ \frac{d^{2}x}{dt^{2}} $$ and that measured in frame S' is $$ \frac{d^{2}x'}{dt'^{2}} $$My question...
I'm working with modfied gravity models and I need to consider the perturbation of field equations. I have problems with the term were I have two covariant derivatives, I'm not sure if I'm doing it right.
I have:
$$\delta(\nabla_\rho \nabla_\nu \left[F'(G)R_{\mu}^{\hphantom{\mu} \rho}\right])$$...
Summary: Failed find information on the internet, really appreciate any help.
Can someone tell me what is ∇ϒ∇δ𝒆β? It seems to be equal to 𝒆μΓμβδ,ϒ+(𝒆νΓνμϒ)Γμβδ. Is this some sort of chain rule or is it by any means called anything?
Calculating the christoffel symbols requires taking the derivatives of the metric tensor. What are you taking derivatives of exactly? Are you taking the derivatives of the inner product of the basis vectors with respect to coordinates? In curvilinear coordinates, for instance curved spacetime in...
I have calculated the normalization constant, but I'm struggling with the discontinuities in the derivatives of the wave function. Due to the symmetry, it should suffice to consider the first two cases. The results should be (according to WolframAlpha):
\left( \frac{\partial^{2}}{\partial...
This was my attempt at a solution and was wondering where did I go wrong: -\frac{\partial}{\partial p_\mu}\frac{1}{\not{p}}=-\frac{\partial}{\partial p_\mu}[\gamma^\nu p_\nu]^{-1}=\gamma^\nu\frac{\partial p_\nu}{\partial p_\mu}[\gamma^\sigma...