? D. 3 x 2 2 y. D Pi r ^ 2 h, r. 4 y. D 3 x ^ 3 2 y ^ 2, y, y. D 4 x 3 y 2 z ^5, z, 2, y, x. This means take partial z first then partial x

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1 PartialsandVectorCalclulus.nb? D D f, x gives the partial derivative f x. D f, x, n gives the multiple derivative n f x n. D f, x, y, differentiates f successively with respect to x, y,. D f, x, x 2, for a scalar f gives the vector derivative f x, f x 2,. D f, array gives a tensor derivative. D Sqrt 3 x ^ 2 2 y, y 3 x 2 2 y D Sqrt 3 x ^ 2 2 y, y. x, y 2 7 D Pi r ^ 2 h, r 2 h Π r D 3 x ^ 3 2 y ^ 2, y 4 y D 3 x ^ 3 2 y ^ 2, y, 2 4 D 3 x ^ 3 2 y ^ 2, y, y 4 D 4 x 3 y 2 z ^5, x, z This means take partial z first then partial x 2 f x z f zx D f,x,z 6 4 x 3 y 2 z 3 D 4 x 3 y 2 z ^5, x, y, z 44 4 x 3 y 2 z 2 D 4 x 3 y 2 z ^5, z, 2, y, x x 3 y 2 z D x ^ 2 y ^ 3 z ^ 4, x, y, z 2 x, 3 y 2, 4 z 3

2 PartialsandVectorCalclulus.nb 2 VectorAnalysis` There are built in coordinate systems each having default variables. The default coordinate system is Cartesian with variables named Xx,Yy,Zz. Default Cylindrical coordinate system variables are named Rr, Ttheta, Zz and default Spherical coordinate system variables are named Rr,Ttheta,Pphi.You can switch to another built in coordinate system. You can also rename the variables in each coordinate system. For example SetCoordinates Cylindrical switches the default system to cylindrical coordinates with the variables listed above. SetCoordinates Cylindrical a,b,c switches to Cylindrical coordinates and changes the names of the variables to a,b,c.? gives the name of the default coordinate system.? SetCoordinates SetCoordinates coordsys sets the default coordinate system to be coordsys with default variables. SetCoordinates coordsys c, c 2, c 3 sets the default coordinate system to be coordsys with variables c, c 2, and c 3.? Coordinates Coordinates gives a list of the default coordinate variables in the default coordinate system. Coordinates coordsys gives a list of the default coordinate variables in the coordinate system coordsys. Coordinates Xx, Yy, Zz Coordinates Cylindrical Rr, Ttheta, Zz Coordinates Spherical Rr, Ttheta, Pphi Cartesian SetCoordinates Cylindrical Cylindrical Rr, Ttheta, Zz Cylindrical

3 PartialsandVectorCalclulus.nb 3? Grad Grad f gives the gradient, f, of the scalar function f in the default coordinate system. Grad f, coordsys gives the gradient of f in the coordinate system coordsys. Grad x ^ 2 z ^ 2 Cos y,, Grad Xx ^ 2 Zz ^ 2 Cos Yy 2 Xx, Sin Yy, 2 Zz Grad x ^ 2 z ^ 2 Cos y, Cartesian x, y, z 2 x, Sin y, 2 z Grad Rr ^ 2 Zz ^ 2 Cos Ttheta,, 2 Zz SetCoordinates Cylindrical Cylindrical Rr, Ttheta, Zz Grad Rr ^ 2 Zz ^ 2 Cos Ttheta 2 Rr, Sin Ttheta, 2 Zz Rr Grad r ^ 2 z ^ 2 Cos Theta, Cylindrical r, Theta, z 2 r, Sin Theta, 2 z r SetCoordinates Spherical R, theta, phi Spherical R, theta, phi Grad R ^ 2 Cos theta Sin phi Sin phi Sin theta 2 R,, R Cos phi Cot theta R

4 PartialsandVectorCalclulus.nb 4? Div Div f gives the divergence, f, of the vector field f in the default coordinate system. Div f, coordsys gives the divergence of f in the coordinate system coordsys. Spherical Div Xx ^ 2, 2 Xx Yy ^ 3, Cos Zz 2 Xx 6 Xx Yy 2 Sin Zz? Curl Curl f gives the curl, f, of the vector field f in the default coordinate system. Curl f, coordsys gives the curl of f in the coordinate system coordsys. Coordinates Xx, Yy, Zz x, y, z Cartesian x, y, z Curl y ^ 3 z x ^ 2, 2 x y ^ 3 z, x Cos z, x 2 y 3 Cos z, 2 y 3 3 x 2 y 2 z Div Curl y ^ 3 z x ^ 2, 2 x y ^ 3 z, x Cos z Grad x ^ 2 z ^ 2 Cos y 2 x, Sin y, 2 z Curl Grad x ^ 2 z ^ 2 Cos y,,

5 PartialsandVectorCalclulus.nb 5? Laplacian Laplacian f gives the Laplacian, 2 f, of the scalar function or vector field f in the default coordinate system. Laplacian f, coordsys gives the Laplacian of f in the coordinate system coordsys. x, y, z Cartesian x, y, z Laplacian f x, y, z f,,2 x, y, z f,2, x, y, z f 2,, x, y, z Laplacian x ^ 2 y ^ 3 z 2 6 y Laplacian x ^ 2 7 y ^ 2 8 z ^ 2 x^2 7y^2 8 z^2 is harmonic FullSimplify Laplacian Log x ^ 2 y ^ 2 2 ArcTan y x Ln x^2 y^2 2 ArcTan y x is harmonic. See below Laplacian f Rr, Ttheta, Zz, Cylindrical Simplify f rr r 2 f Θ r f r f zz f,,2 Rr, Ttheta, Zz f,2, Rr, Ttheta, Zz Rr 2 f,, Rr, Ttheta, Zz f 2,, Rr, Ttheta, Zz Rr Laplacian Log Rr, Cylindrical Ln r is harmonic Laplacian Log Rr 2 Ttheta, Cylindrical Ln r 2 2Θ is harmonic. See above Laplacian f Rr, Ttheta, Pphi, Spherical Rr, Ttheta, Pphi Simplify Rr 2 Csc Ttheta 2 f,,2 Rr, Ttheta, Pphi Cot Ttheta f,, Rr, Ttheta, Pphi f,2, Rr, Ttheta, Pphi 2 Rr f,, Rr, Ttheta, Pphi Rr 2 f 2,, Rr, Ttheta, Pphi