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bigtriangledown

Represents a large downward-pointing triangle operator commonly used in mathematical notation for gradient, divergence, or similar operations.

Overview

Serves as a prominent mathematical operator in advanced calculus, vector analysis, and differential geometry, particularly when working with vector fields and multivariable functions.

  • Often employed to denote the gradient operator in its downward form
  • Appears in physics equations related to potential fields and fluid dynamics
  • Frequently used in tensor calculus and differential operators
  • Can represent a backward difference operator in numerical analysis
  • Sometimes used in theoretical physics to denote certain quantum operators

Examples

Expressing the gradient operator in vector calculus.

▽f(x,y,z)=∂f∂xi+∂f∂yj+∂f∂zk\bigtriangledown f(x,y,z) = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k} 
\bigtriangledown f(x,y,z) = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}

Showing the Laplacian operator in mathematical physics.

▽2ϕ=∂2ϕ∂x2+∂2ϕ∂y2+∂2ϕ∂z2\bigtriangledown^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} 
\bigtriangledown^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}

Representing the divergence of a vector field.

▽⋅F=∂Fx∂x+∂Fy∂y+∂Fz∂z\bigtriangledown \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} 
\bigtriangledown \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}