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The scalar form of Laplace's equation is the partial differential equation del ^2psi=0, (1) where del ^2 is the Laplacian. Note that the operator del ^2 is commonly written as Delta by mathematicians (Krantz 1999, p. 16). Laplace's equation is a special case of the Helmholtz differential equation del ^2psi+k^2psi=0 (2) with k=0, or Poisson's equation del ^2psi=-4pirho (3) with rho=0.
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Show multiple sets on a number line: number line x^2>5, 0<=x<3. Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.Add a comment. 1. Without doing any further computations, you can notice that by addition the first and the third equation you get. (3 − λ)x1 + (3 − λ)x3 = 2. ( 3 − λ) x 1 + ( 3 − λ) x 3 = 2. For λ = 3 λ = 3 this is equation 0 = 2 0 = 2. So you know that for λ = 3 λ = 3 the system has no solution.
The unknown function u is called the dependent variable and is a function of several independent variables like the time t and spatial coordinates like x.Partial differential equations are constructed from terms where the dependent variables are differentiated with respect to the independent variables. For example, in the one-dimensional heat equation , the dependent variable is a function of ...
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