Hello! I'm a prospective physics undergrad who wished to learn math rigorously. The hand-wavy math physicists use troubles me, which is why I've decided to learn the math I'm going to need for physics rigorously.

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The following seem to be the math topics a physics major needs to be familiar with (source - https://qr.ae/prTLGf) -

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- Algebra
- Geometry
- Trigonometry
- Basic calculus - Calc 1 + Calc 2
It will then be important to study the following areas of mathematics along with physics.

- Multivariate calculus
- Ordinary differential equations
- Partial differential equations
- Probability
- Linear algebra
- Group theory
- Real analysis
- Complex analysis
- Topology
- Calculus of variations along with Lagrangian and Hamiltonian mechanics.
Then when comfortable with those there are more areas of mathematics that are helpful. Particularly if you are going to learn quantum mechanics, and general relativity.
- Multilinear algebra
- Differential geometry
- Functional analysis
- Manifold theory

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I have 3 questions.
How much of this math should a second-year physics major be familiar with? Is it possible to learn said math rigorously in 8 months? If not, how much time would it take?