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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/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/Partial_Differential_Equations_(Miersemann)/1%3A_Introduction/1.3%3A_Partial_Differential_Equations155, where h is a continuous function and the associated solution u of the boundary value problem has no finite Dirichlet integral. e., the angle between the container wall and the capillary s...155, where h is a continuous function and the associated solution u of the boundary value problem has no finite Dirichlet integral. e., the angle between the container wall and the capillary surface, defined by v=v(x1,x2), at the boundary. div (Tu) is the left hand side of the minimal surface equation (???) and it is twice the mean curvature of the surface defined by z=u(x1,x2), see an exercise.
- 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/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/Partial_Differential_Equations_(Walet)/05%3A_Separation_of_Variables_on_Rectangular_Domains/5.04%3A_Laplace%E2%80%99s_EquationLaplace's equation are the simplest examples of elliptic partial differential equations. The solutions of Laplace's equation are the harmonic functions, which are important in many fields of science, ...Laplace's equation are the simplest examples of elliptic partial differential equations. The solutions of Laplace's equation are the harmonic functions, which are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. In the study of heat conduction, the Laplace equation is the steady-state heat equation.
- 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.
