Archive of posts in category “math”
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A polynomial whose sum of coefficients is a factorial
The function is a polynomial of degree w.r.t. , and the sum of its coefficients is . This turns out to be properties of Eulerian numbers. -
The point on the circle farthest to two lines
Suppose is a point on the circle . When is the sum of distances from to two edges of extremal? It turns out to be related to angle bisectors (the intersections of and the bisector of or its adjacent supplementary angle are extremals), while the edge cases (at the intersections of and edges of ) are a little tricky: we need to use the bisectors to divide the plane into four quadrants, pick the two quadrants where the line intersecting at lies, translate the region to make it center at , and see whether is inside the translated region. -
Solving linear homogeneous ODE with constant coefficients
By using power series, we can prove that the problem of solving linear homogeneous ODE with constant coefficients can be reduced to the problem of solving a polynomial with those coefficients. This article illustrates this point in detail, but it uses a very awful notation… -
Drawing a heart using Joukowsky transformation
Joukowsky transformation of a circle centered at of radius is a curve resembling a heart. -
Generalization of Euler–Lagrange equation
We may generalize Euler–Lagrange equation to higher dimensional optimization problems: find a function defined inside a region to extremize a functional defined as an integral over that region, with the constraint that the value of the function is fixed on the boundary of the region. -
Normal vectors of a scalar field
This article gives the formula for the normal vectors of a surface defined by a scalar field on . The normal vector of the graph of the function at is . This also provides us a way to recover a scalar field from the normal vectors of its graph: normalizing the vectors so that the last component is , and then integrate the rest components. -
Hyperellipsoids in barycentric coordinates
In this article, I introduce the barycentric coordinates: it is an elegant way to represent geometric shapes related to a simplex. By using it, given a simplex, we can construct a hyperellipsoid with the properties: its surface passes every vertex of the simplex, and its tangent hyperplane at each vertex is parallel to the hyperplane containing all other vertices.
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