“As Feynman diagrams are the most natural language to describe the microscopic process of elementary particles, the graphical notation is the canonical language of the vector calculus system. “As a child playing with educational toys such as Lego blocks or magnetic building sticks, it will be an entertaining experience to ‘doodle with the dancing diagrams,’ they say. Kim and co say their approach changes vector field calculus into a visual task, rather like building with Lego bricks. “The language is highly intuitive and automatically simplifies tensorial expressions,” they say. Kim and co show how their notation turns complex mathematical expressions into relatively simple graphics, just like Feynman diagrams. And they extend their ideas to tensors, which are more complex mathematical objects, each with two or more indices. The researchers go on to describe a wide range of other mathematical tools, such as the del operator along with various important identities used in vector calculus. ![]() The graphic for a cross product is y-shaped, with the lines from the two vectors connecting to a third that extends away. And the results can be fantastically complex-huge, multidimensional matrices.īut a cross product between two vectors produces another vector, and again Kim and co’s notation handles this automatically. Vector fields can be multiplied by scalars or by each other in two different ways, known as a dot product and a cross product. The problems arise when these quantities interact mathematically. A vector can be written as a i where i = 1, 2, or 3 in three-dimensional space. Mathematicians represent these fields using an approach called index notation. So at its very simplest, a vector field can be an infinite list of vectors. The problem is that vector fields are intricate entities-they assign a single vector to every point in three-dimensional space and can themselves be representations of more complex mathematical objects called differentiable manifolds. ![]() ![]() Line integrals, vector integration, physical applications. That’s why every physics and engineering undergraduate spends many happy hours struggling with the mathematics and the arcane notation that it requires. Fields, potentials, grad, div and curl and their physical interpretation, the Laplacian, vector identities involving grad, div, curl and the Laplacian. The reason it is so important in physics is that more or less everything in the universe can be described in terms of vector fields-electromagnetic fields, gravitational fields, fluid flow, and so on. Vector calculus is the branch of mathematics that deals with the differentiation and integration of vector fields.
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