jagomart
digital resources
picture1_Derivatives Calculus Pdf 171462 | Mathematical Physics 04 Partial Derivatives And Vector Calculus


 224x       Filetype PDF       File size 0.63 MB       Source: www.lehman.edu


File: Derivatives Calculus Pdf 171462 | Mathematical Physics 04 Partial Derivatives And Vector Calculus
partial derivatives vector calculus partial derivatives functions of several arguments multivariate functions such as f can be differentiated with respect to each argument f f xf yf x y etc ...

icon picture PDF Filetype PDF | Posted on 26 Jan 2023 | 2 years ago
Partial capture of text on file.
          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
The words contained in this file might help you see if this file matches what you are looking for:

...Partial derivatives vector calculus functions of several arguments multivariate such as f can be differentiated with respect to each argument xf yf x y etc one define higher order the same or different variables for most mixed do not depend on differentiation this holds if are continuous at a given point instance xy bad series taylor generalized and sin cos o in symmetric form normal expand exercise find way sort polynomial increasing powers physics makes use differential operations gradient divergence curl rotor laplacian current version mathematica realizations these new included main body software instead implemented optional vectoranalysis package that has called before performing needs unfortunately seems inconvenient scalar funtion is defined gradf e z speak about operator ex ey ez acts an example force minus potential energy u similar electric field examples trying s vectorcalculus following help clear grad wrong output attempt standard usage d oscillator result still command wo...

no reviews yet
Please Login to review.