Mathematical Review: Partial Differential Equations (PDE), Laplace's Equation, Gradient, Divergence

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In this video I go over a quick math review of the terms involved in the Laplace equation in 3D rectangular coordinates. Laplace's equation is defined as the divergence of the gradient of a function set equal to zero. This is also written using the Laplace Operator or Laplacian (∇² or Δ). The gradient is the vector differential operator which gives a vector of the partial derivatives of a function. Partial derivatives are derivatives of a multivariable function while holding the other variables constant. The divergence is written as the dot product of the del operator with a vector field, and gives a scalar that is the sum of the partial derivatives of the field's components. The divergence of the gradient of a function gives the sum of second partial derivatives. The gradient gives the direction and rate of fastest increase of a scalar function, while the divergence measures the net flux out of a point (or the extent to which a vector field acts as a source or sink at that point).

#math #sphericalharmonics #calculus #partialdifferentialequation #multivariablecalculus

Timestamps

• Laplace's equation is a 2nd order partial differential equation (PDE), which is part of multivariable calculus – 0:00
• Partial derivative is the derivative of a multivariable function while holding the other variables constant – 1:21
• Laplace's equation in rectangular coordinates, and can be written as the divergence of the gradient of f: ∇ · ∇f = ∇^2 f = Δf = Laplacian or Laplace operator – 3:16
• Nabla symbol (upside down triangle ∇) "del", denotes the vector differential operator – 4:40
• Figure: Gradient, ∇f, gives direction and rate of fastest increase of a function f – 6:10
• Divergence is the sum of partial derivatives and uses the dot product as a mnemonic device (∇ ·). Note that Fx, Fy, Fz are vector components of F and not partial derivatives: https://grok.com/share/c2hhcmQtMg_998aa7b7-fe08-49c2-8753-314bab601e49 – 6:49
• Figure: Positive, negative, and zero divergence showing flux (creation / source or sink) or zero flux – 8:06 .

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