Math.vc

Support for vector calculus in JavaScript.

The following methods are accessible through the Math.vc namespace:

• add(A,B)
• sub(A,B)
• mul(A,k)
• dotProduct(A,B)
• crossProduct(A,B)
• outerProduct(A,B)
• kroneckerProduct(A,B)
• angle(A,B)
• projection(A,B)
• arePerpendicular(A,B)
• areParallel(A,B)
• length(V)
• normalize(V)

Plus the following differential methods:

• gradient(f)
• divergence(F)
• curl(F)
• laplacian(f)
• vectorLaplacian(F)

Adds two vectors.

$$ A + B $$

Subtracts two vectors.

$$ A - B $$

Multiplies a vector by a scalar value.

$$ kA $$

The dot product of two vectors.

$$ A \cdot B $$

The cross product of two vectors.

$$ A \times B $$

The outer product of two vectors.

$$ A \otimes B = \begin{bmatrix} a_1b_1 & a_1b_2 & \cdots \\ a_2b_1 & a_2b_2 & \cdots \\ \vdots & \vdots & \ddots \end{bmatrix} $$

The Kronecker product of two vectors.

$$ A \otimes B = [ a_1B, ..., a_nB ] $$

Returns an angle between two vectors.

$$ \arccos\left( \frac{A \cdot B}{|A| |B|} \right) $$

Projects vector A on vector B.

$$ \text{proj}_B \ A = \frac{A \cdot B}{|B|^2} B $$

Checks if two vectors are perpendicular.

$$ A \perp B \iff A \cdot B = 0 $$

Checks if two vectors are parallel.

$$ A \parallel B \iff A \times B = [0,0,0] $$

Returns a length of a vector.

$$ |V| = \sqrt{a_1 + \cdots + a_n} $$

Performs normalization of a vector.

$$ \frac{V}{|V|} $$

• input: $f: \Bbb{R}^3 \to \Bbb{R}$
• output: $\Bbb{R}^3$

$$ \text{grad} \ f = \nabla f = \left[ \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right]^{\intercal} $$

• input: $F: \Bbb{R}^3 \to \Bbb{R}^3$
• output: $\Bbb{R}$

$$ \text{div} \ F = \nabla \cdot F = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} $$

• input: $F: \Bbb{R}^3 \to \Bbb{R}^3$
• output: $\Bbb{R}^3$

$$ \text{curl} \ F = \nabla \times F = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \\ \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ \\ \\ F_x & F_y & F_z \end{vmatrix} $$

• input: $f: \Bbb{R}^3 \to \Bbb{R}$
• output: $\Bbb{R}$

$$ \Delta f = \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} $$

• input: $F: \Bbb{R}^3 \to \Bbb{R}^3$
• output: $\Bbb{R}^3$

$$ \Delta F = \nabla^2 F = (\Delta F_1, \Delta F_2, \Delta F_3) $$

• a scalar field $f: \Bbb{R}^3 \to \Bbb{R}$ receives three arguments ($x,y,z$) and return a scalar value
• a vector field $\Bbb{R}^3$ is an object with the following three properties: dx, dy and dz

To get the following gradient:

$$ \nabla \ f, \text{ for } f(x,y,z) = x^2+y^2+z^2 $$

we shall write:

const f = (x,y,z) => x*x + y*y + z*z;
const grad = Math.vc.gradient(f);

grad(1,2,3)
// {dx: 2, dy: 4, dz: 6 }

To get the following curl:

$$ \text{curl} \ F, \text{ for } F(x,y,z) = (-y,x,0) $$

we shall write:

const F = (x,y,z) => ({
  dx: -y,
  dy: x,
  dz: 0
});
const curl = Math.vc.curl(F);

curl(1,1,1)
// {dx: 0, dy: 0, dz: 2}