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- https://math.libretexts.org/Courses/East_Tennesee_State_University/Book%3A_Differential_Equations_for_Engineers_(Lebl)_Cintron_Copy/4%3A_Fourier_series_and_PDEs/4.9%3A_Steady_state_temperature_and_the_LaplacianSuppose we have an insulated wire, a plate, or a 3-dimensional object. We apply certain fixed temperatures on the ends of the wire, the edges of the plate, or on all sides of the 3-dimensional object....Suppose we have an insulated wire, a plate, or a 3-dimensional object. We apply certain fixed temperatures on the ends of the wire, the edges of the plate, or on all sides of the 3-dimensional object. We wish to find out what is the steady state temperature distribution. That is, we wish to know what will be the temperature after long enough period of time.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Raji)/02%3A_Prime_Numbers/2.03%3A_The_Fundamental_Theorem_of_ArithmeticThe Fundamental Theorem of Arithmetic is one of the most important results in this chapter. It simply says that every positive integer can be written uniquely as a product of primes. The unique factor...The Fundamental Theorem of Arithmetic is one of the most important results in this chapter. It simply says that every positive integer can be written uniquely as a product of primes. The unique factorization is needed to establish much of what comes later. There are systems where unique factorization fails to hold. Many of these examples come from algebraic number theory. We can actually list an easy example where unique factorization fails.
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/15%3A_Fourier_series_and_PDEs/15.09%3A_Steady_State_Temperature_and_the_LaplacianSuppose we have an insulated wire, a plate, or a 3-dimensional object. We apply certain fixed temperatures on the ends of the wire, the edges of the plate, or on all sides of the 3-dimensional object....Suppose we have an insulated wire, a plate, or a 3-dimensional object. We apply certain fixed temperatures on the ends of the wire, the edges of the plate, or on all sides of the 3-dimensional object. We wish to find out what is the steady state temperature distribution. That is, we wish to know what will be the temperature after long enough period of time.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_(Chasnov)/09%3A_Partial_Differential_Equations/9.07%3A_The_Laplace_Equationx=rcosθ,y=rsinθ; and the chain rule gives for the partial derivatives \[\label{eq:4}\frac{\partial u}{\partial r}=\frac{\partial u}{\partial x}\frac{\partial x}{\par...\boldsymbol{\label{eq:3}x=r\cos\theta,\quad y=r\sin\theta ;} and the chain rule gives for the partial derivatives ∂u∂r=∂u∂x∂x∂r+∂u∂y∂y∂r,∂u∂θ=∂u∂x∂x∂θ+∂u∂y∂y∂θ.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/12%3A_Fourier_Solutions_of_Partial_Differential_Equations/12.03%3A_Laplace's_Equation_in_Rectangular_CoordinatesThis section deals with a partial differential equation that arises in steady state problems of heat conduction and potential theory.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/MAT-204%3A_Differential_Equations_for_Science_(Lebl_and_Trench)/06%3A_Fourier_series_and_PDEs/6.09%3A_Steady_State_Temperature_and_the_LaplacianSuppose we have an insulated wire, a plate, or a 3-dimensional object. We apply certain fixed temperatures on the ends of the wire, the edges of the plate, or on all sides of the 3-dimensional object....Suppose we have an insulated wire, a plate, or a 3-dimensional object. We apply certain fixed temperatures on the ends of the wire, the edges of the plate, or on all sides of the 3-dimensional object. We wish to find out what is the steady state temperature distribution. That is, we wish to know what will be the temperature after long enough period of time.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_for_Engineers_(Lebl)/4%3A_Fourier_series_and_PDEs/4.09%3A_Steady_state_temperature_and_the_LaplacianSuppose we have an insulated wire, a plate, or a 3-dimensional object. We apply certain fixed temperatures on the ends of the wire, the edges of the plate, or on all sides of the 3-dimensional object....Suppose we have an insulated wire, a plate, or a 3-dimensional object. We apply certain fixed temperatures on the ends of the wire, the edges of the plate, or on all sides of the 3-dimensional object. We wish to find out what is the steady state temperature distribution. That is, we wish to know what will be the temperature after long enough period of time.
