Math book for semi self-learning

Thread Starter

StrongPenguin

Joined Jun 9, 2018
264
Guess this belongs here, since it is not homework. Yet.

I plan to upgrade my education to bachelors degree, starting in September. My ambition is to do all of this while still sailing, meaning I will miss half of the classes.

So I need good self study books while I am at sea, that work without internet or power. I have seen our math text book, and it did not seem friendly.

Here are the topics we will work through:
Linear equations and linear maps. Matrix algebra. Vektor spaces. Eigenvalue problems. Symmetric and orthogonal matrices. Complex numbers. Linear differential equations. Standard functions. Functions of one and several real variables: linear approximations and partial derivatives, Taylor expansions and quadratic forms, extrema and level curves, line, surface and volume integrals. Vector fields, Gauss' and Stokes' theorem.
Applications of MAPLE in the above areas. Examples of applications in the engineering sciences.
Link to course
https://kurser.dtu.dk/course/2020-2021/01005?menulanguage=dk

Any tips?
 
It looks like that course will be a firehose of techniques for solving problems in linear algebra, differential equations, and multivariable calculus. A potential challenge with such a class is that, though the problem sets may be grounded in an engineering context, the mathematical techniques themselves will seem completely arbitrary and mystifying unless you have a solid background in the topics. This is especially true for linear algebra, which underlies both differential equations and calculus. Unless you're already comfortable with vector spaces and thinking of matrices as linear transformations of a space, I'd highly recommend spending some effort on getting to know the "big picture" of linear algebra.

In other words, don't bother pre-studying how to calculate eigenvectors or a change of basis (you'll learn those in class), rather try to develop some intuition on why we care about such things. There is a holistic aspect to all of the mathematical techniques you'll be learning, and it all stems from linear algebra. You know that the exponential function is its own derivative? In the holistic view, exponentials are eigenvectors of the differential operator on the space of differentiable functions, which is a vector space. The set of solutions to a homogeneous differential equation forms a vector space. A multivariable function defines a scalar field, and the gradient of that function is a vector space. An nth-order Taylor expansion is a change of basis to the vector space of nth-order polynomials. Etc., etc.

I truly wish there were a single book or video one could watch to get an intuitive sense of this deep connection between linear algebra and basically everything else, but I don't think such a thing exists (maybe it's not possible). Short of taking a few linear algebra courses, I suggest searching for and watching high-level videos on linear algebra, maybe browsing through a few books, all while focusing on how the fundamental objects of linear algebra (vectors, spaces, matrices) are actually abstractions of more familiar mathematical objects and, consequently, physical processes.

Best of luck.
 
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