5

If a point is at a unit distance from the origin, and it makes an angle $p$ with the x-axis, then $\cos p$ is defined as the x co-ordinate of that point. $\sin p$ is defined as the y co-ordinate of that point. If a point is a distance $r$ from the origin and makes an angle $p$ wih the x-axis, then its x and y co-ordinates are $r \cos p$ and $r\sin p$ ...


2

Recall the triangle law of vectors. To find the x and y components, construct a triangle with the vector $F_e$ as the hypotenuse. This is where trigonometry comes in- since $F_{ex}$ is the base of the triangle and $F_e$ is the hypotenuse, you can now figure out the trigonometric relation between $\theta$, $F_e$ and $F_{ex}$.


2

So to clarify, you're just trying to plot the vector using it’s x-y components, or or are you asking how to get the $X$ or $Y$ components of a vector? If it’s # 1 it seems like you already did it since there’s no $X$ component and the $Y$ component is in the negative direction. If it’s # 2 then you're probably looking for $x=R\cos\theta$ and $y = R\sin\...


2

Consider the example that Altland suggests: let $\theta,\phi,\psi$ be the Euler angles specifying a rotation. Then $x,y,z \to x(\theta,\phi,\psi),y(\theta,\phi,\psi), z(\theta,\phi,\psi)$ and the derivatives $\partial x/\partial \theta$ etc. give the infinitesimal form of the rotation.


2

Three things I learned from studying these Relativity paradoxes are: The events always happen. If one observer sees "pole hit the back of the barn", so does the other. Most of the "paradox" stems from our intuition of absolute time and / or simultaneity. Two things that are simultaneous in one reference frame need not be simultaneous in ...


1

I was also blocking for a time on this point in the book, I came accross your question which was helpfull for me. In fact, I think the intuitive way is using the derivative of the product of functions: In 3.16, let assume $f$ to be the first term $ \frac{\partial x^{'a}}{\partial x^{d}} $ and $g'$ to be the second $ \frac{{\partial}^2 x^{d}}{\partial x^{'c}\...


1

There are two different ways to do differential geometry: the 'extrinsic' view, and the 'intrinsic' view. The extrinsic view is what you just described: you set up an embedding of your manifold inside a copy of $\mathbb R^n$, and you refer your manifold's geometry to that ambient space. The intrinsic view is what you're asking about: studying the geometry ...


1

First lets clear up some confusion. You say: To be conformal means that the physic is unchanged This is not what "conformal" means. Conformal means that the angles are unchanged. In the context of GR, this in particular means that the causal structure remains unchanged. The phyisics in the meantime can be quite different. In particular, the $uv$-...


Only top voted, non community-wiki answers of a minimum length are eligible