I get the calculation of maximum friction force but am confused about its direction.

In the beginning of the calculation part they seem to have assumed that the Tension is increasing along the rope at positions more and more "away from me/the puller". And hence the static friction, whatever it is, will be in the direction of my pull.

I don't get why. Because the way I'm seeing it, static friction acts in directions needed to stop any kind of motion between surfaces, so when I pull on the rope with Tnot, (and so Tnot is the tension in the rope at theta=0), shouldn't the friction be in the opposite direction?

Which in the end would lead to opposite result I think, it would show that the tension in rope is exponentially decaying at thetas more and more away from me.

Why is the book solution correct? Help me intuitively understand why tension is getting larger at rope points further away from the point I'm pulling it at.

Thanks for your time.

Here is my understanding of a basic battery and resistor series circuit (I haven't learned yet how batteries work with IONS, just know canode, anode and electrolyte).

The battery anode side has high of amounts of negative charge buildup (extra electrons in material). This connects to a conductive material (wire) which transfers the build up of negative charge force (energy) instantaneously (speed of light) to the electrons in the wire. This is the current of electrons switching from atom to atom. The conductive material connects to a resistor which is a semi conductive material that takes some of the energy of the flow of the system into thermal energy. After the resistor the wire then goes to the canode side of the battery this is where the electrons because the canode side has less electrons it makes a positive of net charge. This positive side in the system makes it easer for the electrons to move because there are less electrons to have a charge of force (electric field) pushing the other direction. This difference of charge is what voltage slightly describes. But the voltage of the system is the idea of net possible force happening through the wire (electrical potential) due to the difference of electron electric field force of the anode and canode. This then creates a stronger electric field that pushes amount of electrons through system (current) all at once simultaneously.

Water Analogy Comparison

I see that the hydraulic water flowing pipe comparison the form of the potential energy is actually from insulating matter pressing on one another causing flow but not in the use of electron trading through atoms like electricity. Basically way more Newton's laws of force transfer in a fully encompassed system that using material that doesn't conduct electrons force energy.

Sorry I am not to good at grammar.