Symbolic Mathematic examples in Mathematica

	x = 3; y = 5;
	x + y


	8

Differentiation

	Clear[a, b, x];
	expr = 2 a x + b;
	D[expr, x]


	2 a

Integration

	Integrate[expr, x]

		 2
	b x + a x

Matrix and Vector Operations

	Clear[a, b, c, d, x, y];

	(* matrix *)
	m = {{a, b}, {c, d}}


	{{a, b}, {c, d}}


	(* vector *)
	v = {x, y}


	{x, y}


	(* constant times matrix *)
	k m


	{{a k, b k}, {c k, d k}}


	(* square each element of m *)
	m^2


           2   2     2   2
	{{a , b }, {c , d }}


	(* matrix multiplication *)
	m . m


           2
	{{a  + b c, a b + b d},

                             2
          {a c + c d, b c + d }}


	(* matrix times vector, v acts as column vector *)
	m . v


	{a x + b y, c x + d y}


	(* vector times matrix, v acts as row vector *)
	v . m


	{a x + c y, b x + d y}



	(* transpose matrix *)
	Transpose[m]


	{{a, c}, {b, d}}

Solving Equations

	Clear[x, y]



	Solve[{x + y == 10, 2x - y == 5}, {x, y}]


	{{x -> 5, y -> 5}}


	Solve[{a x^2 + b x + c == 0}, x]


			  2
               -b - Sqrt[b  - 4 a c]
	{{x -> ---------------------},
                        2 a

			  2
               -b + Sqrt[b  - 4 a c]
         {x -> ---------------------}}
			2 a


	a = 1; b = 5; c = 4;
	Solve[{a x^2 + b x + c == 0}, x]


	{{x -> -4}, {x -> -1}}


	b = 10; c = 9;
	Solve[{a x^2 + b x + c == 0}, x]


	{{x -> -9}, {x -> -1}}


	c = 8;
	Solve[{a x^2 + b x + c == 0}, x]


	{{x -> -5 - Sqrt[17]},

          {x -> -5 + Sqrt[17]}}


	N[%]


	{{x -> -9.12311},

          {x -> -0.876894}}