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1. ### What _is_ an indefinite integral?

Given an indefinite integral, \int f(x) dx = F(x) + C, I am having some problems in understanding what this indefinite integral "is". The RHS is clearly a function, but what is the LHS? Judging by the equals sign, it should also be a function, but seemingly it isn't because there's no...
2. ### Elegant solution to this vector equation?

Given are a plane E and a line l in general position. I need to find a plane that contains l and intersects E at a given angle \alpha. All of this happens in R^3. The interesting part is to find the normal of the unknown plane, let us call this normal x. I came up with the following...
3. ### Meaning of A^T A and A A^T

The matrices A^{T}A and AA^{T} come up in a variety of contexts. How should one think about them - is there a way to understand them intuitively, e.g. do they have a geometric interpretation?
4. ### Trouble with function definition

Given is the following function (nevermind what the function h is): g(t, q) = \int_0^1 \frac{\partial h(ts, q)}{\partial(ts)} ds This function is supposed to be defined for t = 0. However, I don't see how - the partial derivative in the integral then becomes \frac{\partial h(0...
5. ### Help with Rolle's theorem

Given is a one-parameter family of planes, through x \cdot n(u) + p(u) = 0 with normal vector n and base point p, both depending on the parameter u. Two planes with parameters u_1 and u_2, with u_1 < u_2, intersect in a line (planes are assumed to be non-parallel). This line also lies...
6. ### Tangent to reparameterized curve

Given is a curve \gamma from \mathbb{R} \rightarrow M for some manifold M. The tangent to \gamma at c is defined as (\gamma_*c)g = \frac{dg \circ {\gamma}}{du}(c) Now, the curve is to be reparameterized so that \tau = \gamma \circ f, with f defining the reparametrization. (f' > 0...
7. ### Solutions of Ax = b (for singular A)

A system of linear equations, Ax = b (with A a square matrix), has a unique solution iff det(A) \ne 0. If b = 0, the system is homogeneous and can be solved using SVD (which gives the null space of A). Now, how can the solution set be characterized for singular A and b \ne 0? If a single...