Local symmetries:

Parallel transport: every location has a fiber, how to relate each fiber? that’s what parallel transport do – relate fibers. note the word parallel generalize “constant”! While in calculus, we know derivative y dot or dy(t)/dt =0 means “constant”. in gauge theory, parallel section or parallel function is that covariant derivative =0, or vanishes!

the deep idea behind is illustrated as below, one in derivative form the other in integral form.

Gauge Transformation: still use currency as example, now we can use various unit bills such as one-dollar bill, 10-dollar-bill or 100-dollar bill, the wealth is preserved but number changes, how to express in math by Gauge transformation?

apply it to the currency example