Calculus Hub
Mathematics // N-Dimensional Space

MULTIVARIATE
CALCULUS

Single-variable calculus assumes the world moves on a straight line. Multivariate calculus embraces the reality of 3D space. It is the language of mountains, valleys, vector fields, and multidimensional optimization.

Partial Derivatives

If you are standing on the side of a mountain, your slope depends entirely on which direction you step. A partial derivative measures the rate of change in one specific direction, holding all other variables perfectly still.

f(x,y)=sin⁡(x)⋅sin⁡(y)f(x, y) = \sin(x) \cdot \sin(y)

The notation uses the curly "del" symbol (∂\partial). If we want to find the slope moving strictly along the X-axis, we treat yy as a constant:

∂f∂x=cos⁡(x)⋅sin⁡(y)\frac{\partial f}{\partial x} = \cos(x) \cdot \sin(y)

The Gradient Field

If you calculate the partial derivatives for every dimension, you can bundle them together into a single vector called the Gradient (∇\nabla).

∇f=⟨∂f∂x,∂f∂y⟩\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right\rangle
The Gradient Vector is pure magic. No matter where you are standing on a surface, the gradient vector will always point in the direction of the steepest possible ascent. This mathematical property is the entire foundation of how AI neural networks "learn" via Gradient Descent.

Topology Visualizer

3D Isomorphic Projection

f(x,y) = x² - y²

Vector Field Simulator

f(x,y) = sin(x)sin(y)

Click to drop tracer

Gradient Ascent: You are looking down at a 2D map of a bumpy 3D surface. The red arrows are the Gradient (∇f), pointing toward the steepest uphill incline. Drop a green particle to watch it follow the math to the nearest local maximum!

Advanced Calculus

Initialize theoretical modules for higher-dimensional topologies.

Vector Calculus

Divergence, Curl, and the math that powers electromagnetism and fluid dynamics.

Lagrange Multipliers

Finding the absolute maximum or minimum of a system subject to strict constraints.

Multiple Integration

Calculating absolute volume under 3D surfaces and mass centers of solid objects.