Need to understand the equations in calculus

[Shourya Tripathi]

Homework Statement

I was in the middle of a proof untill I came up with an equation that came out of nowhere and now I am not able to think how that came. So I need help with the understanding of the following equations and how we came up with it. It actually is an attempt to show that on changing the components position vector R, the components of grad T(temperature as an example of scalar field varying with respect to position) change appropriately.

Relevant Equations

x' = x.cos(a) + y.sin(b) ;
y' = -x.sin(a) + y.cos(a);

 

[PeroK]

The text involves a standard rotation of the axes by an angle $\theta$. I'm not sure what your question is.

 

[BvU]

Homework Statement: I was in the middle of a proof until I came up with an equation that came out of nowhere

Which equation ?

and now I am not able to think how that came. So I need help with the understanding of the following equations and how we came up with it. It actually is an attempt to show that on changing the components position vector R, the components of grad T(temperature as an example of scalar field varying with respect to position) change appropriately.

!
A far cry indeed from curvature of space-time

TL;DR Summary: Since it seems so non intuitive that due to its curvature, the space tends to push us towards that mass. I thought that there could be something that is responsible for that, since how can nothing push something?

I don't see no $a$ and $b$; do you mean

Relevant Equations: $$\begin{align*}x' &= \phantom - x\cos\theta + y\sin\theta \tag {2.16}\\ y' &= -x\sin\theta + y\cos\theta \tag {2.17}\end{align*}$$

My neck hurts

And a reference to an even clearer picture with a more complete context here wouldn't harm either

$\$

 

[Shourya Tripathi]

The text involves a standard rotation of the axes by an angle $\theta$. I'm not sure what your question is.

I just need a brief explanation on it and want to understand how we made our approach to that equation.

 

[PeroK]

I just need a brief explanation on it and want to understand how we made our approach to that equation.

Just do an Internet search for rotation of coordinates or coordinate axes.

 

[WWGD]

I just need a brief explanation on it and want to understand how we made our approach to that equation.

That looks like a rotation of the plane. It's straightforward, a linear map, thus representable and, in your case, reptesented, as a matrix. Look up " rotation matrix in R^2", or " Rotation of the Plane".

 

[FactChecker]

I just need a brief explanation on it and want to understand how we made our approach to that equation.

It's not calculus or anything higher than algebra. Just look at diagram (a), especially for simple cases of $\theta=$0 deg, 45 deg, 90 deg, 180 deg. and you will see that the equations are correct.

 

[vela]

The simplest derivation uses basic trig.

https://en.wikipedia.org/wiki/Rotation_of_axes_in_two_dimensions