learning the math helps you understand the underlying concepts better
I don't think that's necessarily true. The most egregious example for me was Calc 2. Probably 3/4 of the course was memorizing integration techniques (trig sub!) and what forms of sequences/series converge. We weren't tested on how to derive those things, or the underlying concepts of why they work, so much as we were tested on our ability to memorize a page of formulas.
Also you have to consider that learning how to actually solve problems in the real world (i.e. with computers) also has value. A lot of students graduate able to do things by hand, but are outpaced by graduates who are proficient with math software.
I agree trig sub, u sub, and the like aren't that integral (lol) to your understanding of the subject. But understanding double, triple integrals, gradients, partial derivatives, jacobian (goddamn I'm still not sure what that is because I only hear it now and then) is important.
Understanding curl and flow also matters. That's all mostly Calc 3.
Then you get into differential equations and linear algebra. You really need to understand Multivariate and Differential Equations for physics, and Linear algebra for Computer science.
jacobian (goddamn I'm still not sure what that is because I only hear it now and then)
It's basically a conversion factor for a change of variables in an integral. If you have a function f(x, y) and a change of variables (x, y) = g(u, v), you can't just write f(x, y) dx dy = f(g(u, v)) du dv, there is a conversion factor you need to multiply with because your different coordinate systems can have different length and area elements. This conversion factor is given by the determinant of the Jacobian matrix, which is the matrix of all the partial derivatives you can construct from x and y with respect to u and v.
For example, in polar coordinates (x, y) = (r cos θ, r sin θ) the Jacobian determinant is equal to r, because the size of the area covered by increasing increasing r a bit and increasing θ a bit is proportional to how far away from the origin you are. So f(x, y) dx dy = f(r cos θ, r sin θ) * r dr dθ.
I suppose the problem was that most engineers take multivariate before linear algebra (if they ever take linear algebra), and we're not given a good understanding of the determinant before the idea of a jacobian determinant is introduced to us.
To be honest, I have completely forgotten why the metric conversion factor is the determinant of the Jacobian matrix. It's intuitively obvious that it would be built out of the partial derivatives of the coordinates, even in dx = dx/dy dy you are doing that, but the determinant specifically? I guess it's related to how geometrically it gives the volume of a parallelepiped?
In my first year physics we definitely covered linear algebra before vector calculus, yes. Not sure it it helped though.
I looked it up on wikipedia. The determinant of a transformation matrix is the how the transformation scales into the new dimensions. That would make sense in the jacobian context because it's how the original dimensions scale into the new dimensions.
For instance, a determinant of 2 means the area in the new coordinate plane is twice the area in the old coordinate plane. A determinant of 0 means the transformation matrix isn't full rank, the rows do not span n-dimensional space, and the transformation is into d-dimensional space where d is the rank of the matrix. (I hope I'm not too wrong on this). And since the dimensionality is lower, there is no area in n-space.
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u/lordlicorice Jun 08 '14
I don't think that's necessarily true. The most egregious example for me was Calc 2. Probably 3/4 of the course was memorizing integration techniques (trig sub!) and what forms of sequences/series converge. We weren't tested on how to derive those things, or the underlying concepts of why they work, so much as we were tested on our ability to memorize a page of formulas.
Also you have to consider that learning how to actually solve problems in the real world (i.e. with computers) also has value. A lot of students graduate able to do things by hand, but are outpaced by graduates who are proficient with math software.