High School Classes for Engineering

What math classes will be needed for me to be able to major in engineering in a school like University of Michigan?
would multivariable calc be needed and linear algebra be needed?

Nope. For sure take AP Calc AB (and BC if you can). STEM focused high schools sometimes offer more advanced math and although that would be great in order to show interest in math, it is by no means required (since it isn’t available in most high schools).

So I’ll be a junior taking brief calc then calc bc senior year, is there a way i could somehow get a year ahead? meaning could i take those 2 in a single year or take BC and Multivariable together?

Depends on your school’s regulations. I wouldn’t reccoment taking multi and BC at the same time though. Don’t feel pressured to take advanced math unless you feel the math is easy for you and you want to go more in depth. I’m going to an engineering school next year and have only taken BC.

As a junior, will you have completed precalculus and trigonometry? If so, why not just take BC as a junior?

BC or equivalent (college calculus 1 and 2) is normally a prerequisite, not a corequisite, for multivariable calculus. Also, most schools do not even require calculus in high school (much less multivariable calculus) for engineering majors (but you have to be ready for calculus at least). For example, Michigan’s mechanical engineering schedule template starts with calculus 1 (Math 115) in the first semester: http://me.engin.umich.edu/sites/default/files/ME%20Degree%20Flow%20Chart%202013%20v1.2.jpg

If you can take MVC, there’s no reason not to. I wouldn’t recommend taking it and BC at the same time, though, especially once you get to some of the nastier 3d integration techniques.

Be careful of getting too far ahead. The quality of those classes matters more than their title. They aren’t check boxes to say you’ve completed them, but rather very foundational to the classes that follow. Sometimes the high school equivalents aren’t up to par to replace a college level course. Plus you won’t be rocketing ahead anyway as physics will be your rate limiter. I wouldn’t see an advantage of going past BC unless you can skip AB (many schools you can’t) AND the instruction is of very high quality.

Fact: When I was in high school, the most advanced math I took was AP Calculus AB. That didn’t stop me from getting undergraduate and graduate degrees from “top 10” programs. Don’t worry so much about getting as far as you possibly can in math. Learn the topics. They’re important later.