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Partial derivatives & Vector calculus Partial derivatives Functions of several arguments (multivariate functions) such as f[x,y] can be differentiated with respect to each argument ∂f ∂f ≡ ∂xf, ≡ ∂yf, ∂x ∂y etc. One can define higher-order derivatives with respect to the same or different variables ∂2f ∂2f ∂2f ∂ ∂f ≡ ∂x,xf, ≡ ∂y,yf, ≡ ≡ ∂x,yf ∂x2 ∂y2 ∂x∂y ∂x ∂y For most of the functions mixed partial derivatives do not depend on the order of differentiation ∂2f ∂2f ≡ ∂x∂y ∂y∂x This holds if the mixed derivatives are continuous at a given point. For instance, fx_, y_ = xy; ∂xfx, y ∂yfx, y ∂y,xfx, y ∂x,yfx, y y x 1 1 "Bad" functions Multivariate series Taylor series can be generalized for multivariate functions and the fx_, y_ = Sinx+y; Seriesfx, y, x, 0, 3 1 1 Siny+Cosyx− Sinyx2− Cosyx3+Ox4 2 6 Seriesfx, y, x, 0, 3, y, 0, 3 y3 y2 y y3 1 y2 y− +Oy4 + 1− +Oy4 x+ − + +Oy4 x2+ − + +Oy4 x3+Ox4 6 2 2 12 6 12 or, in the symmetric form NormalSeriesfx, y, x, 0, 3, y, 0, 3 ExpandNormalSeriesfx, y, x, 0, 3, y, 0, 3 x3 x2 x x3 1 x2 x− + 1− y+ − + y2 + − + y3 6 2 2 12 6 12 x3 x2 y xy2 x3 y2 y3 x2 y3 x− +y− − + − + 6 2 2 12 6 12 Exercise: Find a way to sort this polynomial in increasing powers of x, y. Vector calculus Physics makes use of vector differential operations on functions such as gradient, divergence, curl (rotor), Laplacian, etc. In the current version of Mathematica realizations of these operations are new and not included in the main body of the software. Instead, these functions are implemented in the optional VectorAnalysis package that has to be called before performing these operations Needs"VectorAnalysis`" Unfortunately, this package seems to be inconvenient. Gradient Gradient of a scalar funtion is a vector defined as gradf≡∇f≡e ∂f +e ∂f +e ∂f ≡∂ f,∂ f,∂ f x ∂x y ∂y z ∂z x y z One can speak about the gradient operator defined as ∂ ∂ ∂ ∇ ≡ ex +ey +ez ∂x ∂y ∂z that acts on scalar functions of vector arguments. An example of a gradient in physics is force F that is minus gradient of the potential energy U[x,y,z] and similar for the electric field E that is minus gradient of the electric potential f F ≡ −∇U, E ≡ −∇φ Examples trying to use the Mathematica's VectorCalculus package: Following Mathematica help: Clearx, y, z, U 2 2 2 U = x +y +z ; GradU 0, 0, 0 - a wrong output. An attempt of a standard usage Ux_, y_, z_ = x2 +y2 +z2; ∗ 3d oscillator ∗ Fx_, y_, z_ = GradUx, y, z 0, 0, 0 - same wrong result. Still this command is working with a special naming choice GradXx2+Yy2+Zz2 2Xx, 2Yy, 2Zz However, this naming restriction is inconvenient. Fortunately, it is not difficult to program the gradient in Mathematica. To use the definition below, quit the kernel to remove the VectorAnalysis package from the memory In[43]:= Gradf_ := ∂xf, ∂yf, ∂zf Since x,y,z enters the definition of this function, the arguments of f should also be x,y,z. With any other notation for the arguments of f, it won't work, in contrast to definitions of tru functions. MyGrad works on expressions Gradx2+y2+z2 ∗ ∇r2=2r ∗ 2x, 2y, 2z This means ∇r2 = 2 r MyGrad also works on functions Ux_, y_, z_ = x2 +y2 +z2; ∗ 3d oscillator ∗ Vx_, y_, z_ = x4; GradUx, y, z GradVx, y, z 2x, 2y, 2z 4x3, 0, 0 Mathematica has the symbol “ but it seems it is only for typing ?∇ “ à ?E e à I True One can define a vector function that is the gradient of a scalar function. For the electric field of a point charge one has φx_, y_, z_ = kQ ; ∗ Coulomb potential of a point charge ∗ x2 +y2 +z2 EEx_, y_, z_ = −Gradφx, y, z ∗ Electric field of a point charge ∗ kQx 3 2, kQy 3 2, kQz 3 2 x2 +y2 +z2 x2 +y2 +z2 x2 +y2 +z2 Divergence Divergence of a vector is a scalar defined by ∂Ax ∂Ay ∂Az div A ≡ ∇ ⋅A ≡ + + ∂x ∂y ∂z divergence can be represented by the operator ∂ ∂ ∂ ∇ ≡ ex +ey +ez ∂x ∂y ∂z same as the gradient operator above. The only difference between them is that gradient acts on scalars and divergence acts on vectors. In[61]:= DivAvec_ := ∂xAvec1+∂yAvec2+∂zAvec3 Examples Divx, y, z 3 Fvecx_, y_, z_ = x2, y2, z2 ; DivFvecx, y, z 2x+2y+2z Fvecx_, y_, z_ = y, x, xy; DivFvecx, y, z 0 Laplacian Laplacian is a second-order vector differential operation. Laplacian of a scalar f is defined as div grad f and denoted by D or 2 “ ∆f ≡ divgradf ≡ ∇ ⋅∇ f ≡ ∇2f From this definition follows ∂2f ∂2f ∂2f ∆f = + + ∂x2 ∂y2 ∂z2 The Laplace operator ∂2 ∂2 ∂2 ∆ ≡ ∇2= + + ∂x2 ∂y2 ∂z2 can be obtained by squaring the gradient / divergence operator above. Laplacef_ := ∂x,xf +∂y,yf +∂z,zf Example
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