Content
- inverse equation
- polynomial equation
- Algebra_equation
- 2D equations
- Diophantine equation
- system of equations
- 2D parametric equations
- 3D equations
- 3D parametric equations
- 4D equations
- congruence equation
- Modulus equation
- Probability_equation
- recurrence_equation
- functional_equation
- difference_equation
- vector_equation
- Inequalities
- differential equation
- fractional differential equation
- system of differential equations
- partial differental equation
- integral equation
- fractional integral equation
- differential integral equation
- test solution
- and
- list
- Interactive plot 互动制图
- parametric plot, polar plot
- solve equation graphically
area plot with integral - complex plot
- Geometry 几何
- plane graph 平面图 with plot2D
- function plot with funplot
- differentiate graphically with diff2D
- integrate graphically with integrate2D
- solve ODE graphically with odeplot
- surface in 3D with plot3D
- contour in 3D with contour3D
- wireframe in 3D with wirefram3D
- complex function in 3D with complex3D
- a line in 3D with parametric3D
- a column in 3D with parametric3D
- the 4-dimensional object (x,y,z,t) in 3D with implicit3D
Do exercise and learn from example.
- Fraction `1E2-1/2`
- Add prefix "big" to number for Big number:
big1234567890123456789 - mod operation:
input mod(3,2) for 3 mod 2Complex 复数
math handbook chapter 1.1.2 complexComplex(1,2) number is special vector, i.e. the 2-dimentional vector, so it can be operated and plotted as vector.
- complex numbers in the complex plane:
complex(1,2) = 1+2i - input complex number in polar(r,theta*degree) coordinates:
polar(1,45degree) - input complex number in polar(r,theta) coordinates for degree by polard(r,degree):
polard(1,45) - input complex number in r*cis(theta*degree) format:
2cis(45degree) - in order to auto plot complex number as vector, input complex(1,-2) for 1-2i,
or convert 1-2i to complex(1,-2) by
convert(1-2i to complex) = tocomplex(1-2i) - input complex number in polar:
tocomplex(polar(1,45degree)) - Convert complex a+b*i to polar(r,theta) coordinates:
convert 1-i to polar = topolar(1-i) - Convert complex a+b*i to polar(r,theta*degree) coordinates:
topolard(1-i) - complex 2D plot:
complex2D(x^x)more are in complex2D
- complex 3D plot:
complex3D(pow(x,x))more are in complex function
Numerical approximations
- numeric computation end with the equal sign =
sin(pi/4)= - numeric computation with the ≈ button :
sin(pi/4) - numeric computation with the ≈≈ button :
sin(pi/4) - Convert back with numeric computation n( ) :
n(polar(2,45degree))
n( sin(pi/4) )
n( sin(30 degree) ) - `sin^((0.5))(1)` is the 0.5 order derivative of sin(x) at x=1 :
n( sin(0.5,1) ) - `sin(1)^(0.5)` is the 0.5 power of sin(x) at x=1 :
n( sin(1)^0.5 ) - simplify:
taylor( (x^2 - 1)/(x-1) ) - expand:
expand( (x-1)^3 ) - factorization:
factor( x^4-1 ) - factorizing:
factor( x^2+3*x+2 ) - tangent equation at x=1
tangent( sin(x),x=1 ) - tangentplot( ) show dynamic tangent line when your mouse over the curve.
tangentplot( sin(x) )convert
convert( sin(x) to exp) is the same as toexp(sin(x)) - convert to exp:
toexp( cos(x) ) - convert to trig:
convert exp(x) to trig - convert sin(x) to exp(x):
convert sin(x) to exp = toexp( sin(x) ) - Convert to exp(x):
toexp(Gamma(2,x)) - input sin(x), click the inverse button
inverse( sin(x) )
check its result by clicking the inverse button again.
In order to show multi-value, use the inverse equation instead function.inverse equation
- inverse equation to show multivalue if it has:
inverse( sin(x)=y )
check its result by clicking the inverse button again.polynomial
math handbook chapter 20.5 polynomial - the unit polynomial:
Enter poly(3,x) = poly(3) for the unit polynomial with degree 3: x^3+x^2+x+1. - Hermite polynomial:
hermite(3,x) gives the Hermite polynomial while hermite(3) gives Hermite number. - harmonic polynomial:
harmonic(-3,1,x) = harmonic(-3,x) - the zeta polynomial:
zeta(-3,x) - simplify:
taylor( (x^2 - 1)/(x-1) ) - expand polynomial:
expand(hermite(3,x)) - topoly( ) convert polynomial to
polys( ) as holder of polynomial coefficients,
convert `x^2-5*x+6` to poly = topoly( `x^2-5*x+6` ) - simplify polys( ) to polynomial:
simplify( polys(1,-5,6,x) ) - polyroots( ) is holder of polynomial roots,
topolyroot( ) convert a polynomial to polyroots.
convert (x^2-1) to polyroot = topolyroot(x^2-1) - polysolve( ) numerically solve polynomial for multi-roots:
polysolve(x^2-1) - nsolve( ) numerically solve for a single root:
nsolve(x^2-1) - solve( ) for sybmbloic and numeric roots:
solve(x^2-1) - construct polynomial from roots, activate polyroots( ) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify( ) button.
simplify( polyroots(2,3) )Number
When the variable x of polynomial is numnber, it becomes polynomial number, please see Number_Theory section. - expand Trigonometry by expandtrig( ) :
expandtrig( sin(x)^2 ) - inverse function :
inverse( sin(x) ) - plot a multivalue function by the inverse equation :
inverse( sin(x)=y ) - expand trig function :
expand( sin(x)^2 ) - expand special function :
expand( gamma(2,x) ) - factor :
factor( sin(x)*cos(x) ) - complex 2D plot :
complex2D(x^x)more are in complex2D
- complex 3D plot :
complex3D(pow(x,x))more are in complex function
special Function
math handbook chapter 12 special Function - click the lim( ) button for Limit at x->0 :
`lim_(x->0) sin(x)/x ` = lim sin(x)/x as x->0 = lim(sin(x)/x) - click the nlim( ) button for numeric limit at x->0 :
nlim(sin(x)/x) - click the limoo( ) button for Limit at x->oo :
`lim _(x->oo) log(x)/x` = lim( log(x)/x as x->inf )
= limoo( log(x)/x )
- one side limit, left or right side :
lim(exp(-x),x,0,right)Derivatives
Math Handbook chapter 5 differential calculus - Differentiate
`d/dx sin(x)` = d(sin(x)) - Second order derivative :
`d^2/dx^2 sin(x)` = d(sin(x),x,2) = d(sin(x) as x order 2) - sin(0.5,x) is inert holder of the 0.5 order derivative `sin^((0.5))(x)`, it can be activated by simplify( ):
simplify( sin(0.5,x) ) - Derivative as x=1 :
`d/dx | _(x->1) x^6` = d( x^6 as x->1 ) - Second order derivative as x=1 :
`d^2/dx^2| _(x->1) x^6` = d(x^6 as x->1 order 2) = d(x^6, x->1, 2)Fractional calculus
Fractional calculus - semiderivative :
`d^(0.5)/dx^(0.5) sin(x)` = d(sin(x),x,0.5) = d( sin(x) as x order 0.5) = semid(sin(x)) - input sin(0.5,x) as the 0.5 order derivative of sin(x) for
`sin^((0.5))(x)` = `sin^((0.5))(x)` = sin(0.5,x) - simplify sin(0.5,x) as the 0.5 order derivative of sin(x) :
`sin^((0.5))(x)` = simplify(sin(0.5,x)) - 0.5 order derivative again :
`d^(0.5)/dx^(0.5) d^(0.5)/dx^(0.5) sin(x)` = d(d(sin(x),x,0.5),x,0.5) - Minus order derivative :
`d^(-0.5)/dx^(-0.5) sin(x)` = d(sin(x),x,-0.5) - inverse the 0.5 order derivative of sin(x) function :
f(-1)( sin(0.5)(x) ) = inverse(sin(0.5,x)) - Derive the product rule :
`d/dx (f(x)*g(x)*h(x))` = d(f(x)*g(x)*h(x)) - … as well as the quotient rule :
`d/dx f(x)/g(x)` = d(f(x)/g(x)) - for derivatives :
`d/dx ((sin(x)* x^2)/(1 + tan(cot(x))))` = d((sin(x)* x^2)/(1 + tan(cot(x)))) - Multiple ways to derive functions :
`d/dy cot(x*y)` = d(cot(x*y) ,y) - Implicit derivatives, too :
`d/dx (y(x)^2 - 5*sin(x))` = d(y(x)^2 - 5*sin(x)) - the nth derivative formula :
` d^n/dx^n (sin(x)*exp(x)) ` = nthd(sin(x)*exp(x)) - indefinite integrate :
`int` sin(x) dx
= integrate(sin(x))
- enter a function sin(x), then click the ∫ button to integrate :
`int(cos(x)*e^x+sin(x)*e^x)\ dx` = int(cos(x)*e^x+sin(x)*e^x)
`int tan(x)\ dx` = integrate tan(x) = int(tan(x)) - Exact answers for integral :
`int (2x+3)^7` dx = int (2x+3)^7 - Multiple integrate :
`int int (x + y)\ dx dy` = int( int(x+y, x),y)
`int int exp(-x)\ dx dx` = integrate(exp(-x) as x order 2) - Definite integration :
`int _1^3` (2*x + 1) dx = int(2x+1,x,1,3) = int(2x+1 as x from 1 to 3) - Improper integral :
`int _0^(pi/2)` tan(x) dx =int(tan(x),x,0,pi/2) - Infinite integral :
`int _0^oo 1/(x^2 + 1)` dx = int(1/x^2+1),x,0,oo) - Definite integration :
`int_0^1` sin(x) dx = integrate( sin(x),x,0,1 ) = integrate sin(x) as x from 0 to 1integrator
If integrate( ) cannot do, please try integrator(x) : - integrator(sin(x))
- enter sin(x), then click the ∫ dx button to integrator
fractional integrate
- semi integrate, semiint( ) :
`int sin(x) \ dx^(1/2)` = int(sin(x),x,1/2) = int sin(x) as x order 1/2 = semiint(sin(x)) = d(sin(x),x,-1/2) - indefinite semiintegrate :
`int sin(x)\ dx^0.5` = `d^(-0.5)/dx^(-0.5) sin(x)` = int(sin(x),x,0.5) = semiint(sin(x)) - Definite fractional integration :
`int_0^1` sin(x) `(dx)^0.5` = integrate( sin(x),x,0.5,0,1 ) = semiintegrate sin(x) as x from 0 to 1numeric computation
- numeric computation by click on the "~=" button :
n( `int _0^1` sin(x) dx )numeric integrate
If numeric computation ail, please try numeric integrate nintegrate( ) or nint( ) :
nint(sin(x),x,0,1) = nint(sin(x))integrate graphically
some functions cannot be differentiated or integrated symbolically, but can be semi-differentiated and integrated graphically in integrate2D. - inverse an equation to show multivalue curve.
inverse( sin(x)=y )
check its result by clicking the inverse button again.polynomial equation
- polyroots( ) is holder of polynomial roots, topolyroot( ) convert a polynomial to polyroots.
convert (x^2-1) to polyroot = topolyroot(x^2-1) - polysolve( ) numerically solve polynomial for multi-roots.
polysolve(x^2-1) - construct polynomial from roots, activate polyroots( ) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify( ) button.
simplify( polyroots(2,3) ) - solve( ) for sybmbloic and numeric roots :
solve(x^2-1)
solve( x^2-5*x-6 ) - solve equation and inequalities, by default, equation = 0 for default unknown x if the unknown omit.
solve( x^2+3*x+2 ) - Symbolic roots :
solve( x^2 + 4*x + a ) - Complex roots :
solve( x^2 + 4*x + 181 ) - solve equation for x.
solve( x^2-5*x-6=0,x ) - numerically root :
nsolve( x^3 + 4*x + 181 ) - nsolve( ) numerically solve for a single root.
nsolve(x^2-1)Algebra Equation f(x)=0
math handbook chapter 3 algebaic Equationsolve( ) also solve other algebra equation, e.g. exp( ) equation,
- Solve nonlinear equations:
solve(exp(x)+exp(-x)=4)2D equations f(x,y) = 0
One 2D equation for 2 unknowns x and y, f(x,y) = 0, solved graphically by implicitplot( ) - solve x^2-y^2=1 graphically
x^2-y^2-1=0congruence equation
a x ≡ b (mod m)math handbook chapter 20.3 congruence
- By definition of congruence, a x ≡ b (mod m) if a x − b is divisible by m. Hence, a x ≡ b (mod m) if a x − b = m y, for some integer y.
Rearranging the equation to the equivalent form of Diophantine equation a x − m y = b :
x^2+3x+2=1*(mod 11)
x^2+3x+2=1 mod(11)Modulus equation
- solve( ) Modulus equation for the unknown x inside the mod( ) function :
input mod(x,2)=1 for
x mod 2 = 1
click the solve button - Enter mod(x^2-5x+7,2)=1 for
(x^2-5x+7) mod 2 = 1 - Enter mod(x^2-5x+6,2)=0 for
(x^2-5x+6) mod 2 = 0Diophantine equation f(x,y) = 0
math handbook chapter 20.5 polynomialIt is that number of equation is less than number of the unknown, e.g. one equation with 2 unknowns x and y.
- One 2D equation f(x,y) = 0 for 2 integer solutions x and y
solve( 3x-2y-2=0, x,y )
solve( x^2-3x-2y-2=0, x,y )system of equations f(x,y)=0, g(x,y)=0
math handbook chanpter 4.3 system of equations - system of 2 equations f(x,y)=0, g(x,y)=0 for 2 unknowns x and y by default if the unknowns omit with the solve() button :
solve( 2x+3y-1=0, 3x+2y-1=0 ) - system of 2 equations f(x,y)=0 and g(x,y)=0 for 2 unknowns x and y by default if the unknowns omit.
On First graph it is solved graphically, where their cross is solution:
( 2x+3y-1=0 and 3x+2y-1=0 )2D parametric equations x=f(t), y=g(t)
A system of 2 equations with a parameter t for 2 unknowns x and y, x=f(t), y=g(t), solved graphically : - parametricplot( x=cos(t), y=sin(t) )
- parametric3D( cos(t),sin(t) )
- parametric2D( cos(t),sin(t) )
2D parametric equations x=f(u,v), y=f(u,v)
A system of 3 equations with 2 parameters u and v for 3 unknowns x and y and z, x=f(u,v), y=f(u,v) solved graphically : - parametric3D( cos(u*v),sin(u*v),u*v )
- wireframe3D( cos(x*y),sin(x*y) )
- parametric3D( cos(t),sin(t) )
- 2D surface
3D equations
3D parametric equations x=f(t), y=f(t), z=f(t)
A system of 3 equations with a parameter t for 3 unknowns x and y and z, x=f(t), y=f(t), z=f(t), solved graphically : - parametric3D( t,cos(t),sin(t) )
3D parametric equations x=f(u,v), y=f(u,v), z=f(u,v)
A system of 3 equations with 2 parameters u and v for 3 unknowns x and y and z, x=f(u,v), y=f(u,v), z=f(u,v), solved graphically : - parametric3D( u,u-v,u*v )
- parametric surface
One 3D equation f(x,y,z) = 0
One equation for 3 unknowns x and y and z, f(x,y,z) = 0, solved graphically : - implicit3D( x-y-z )
- plot3D( x-y-z )
4D equations
One 4D equation with 4 variables, - f(x,y,z,t) = 0, solved graphically :
implicit3D( x-y-z-t ) - f(x,y,n,t) = 0, solved graphically :
plot2D( x-y-n-t )Probability_equation
- solve( ) Probability equation for the unknown k inside the Probability function P( ),
solve( P(x>k)=0.2, k)recurrence_equation
- rsolve( ) recurrence and functional and difference equation for y(x)
y(x+1)=y(x)+x
y(x+1)=y(x)+1/x - fsolve( ) recurrence and functional and difference equation for f(x)
f(x+1)=f(x)+x
f(x+1)=f(x)+1/x
functional_equation
- rsolve( ) recurrence and functional and difference equation for y(x)
y(a+b)=y(a)*y(b)
y(a*b)=y(a)+y(b) - fsolve( ) recurrence and functional and difference equation for f(x)
f(a+b)=f(a)*f(b)
f(a*b)=f(a)+f(b)
difference equation
- rsolve( ) recurrence and functional and difference equation for y(x)
y(x+1)-y(x)=x
y(x+2)-y(x+1)-y(x)=0 - fsolve( ) recurrence and functional and difference equation for f(x)
f(x+1)-f(x)=x
f(x+2)- f(x+1)-f(x)=0
vector equation
see vectorInequalities
- solve( ) Inequalities for x.
solve( 2*x-1>0 )
solve( x^2+3*x+2>0 )differential equation
Math handbook chapter 13 differential equation.
ODE( ) and dsolve( ) and lasove( ) solve ordinary differential equation (ODE) for unknown y. - Solve linear ordinary differential equations:
y'=x*y+x
y'= 2y
y'-y-1=0 - Solve nonlinear ordinary differential equations:
(y')^2-2y^2-4y-2=0
dsolve( y' = sin(x-y) )
dsolve( y(1,x)=cos(x-y) )
dsolve( ds(y)=tan(x-y) ) - 2000 examples of Ordinary differential equation (ODE)
more examples in bugs
solve graphically
The odeplot( ) can be used to visualize individual functions, First and Second order Ordinary Differential Equation over the indicated domain. Input the right hand side of Ordinary Differential Equations, y"=f(x,y,z), where z for y', then click the checkbox. by default it is first order ODE. - second order ODE
y''=y'-yintegral equation
Math handbook chapter 15 integral equation.indefinite integral equation
- indefinite integral equation
input ints(y) -2y = exp(x) for
`int y` dx - 2y = exp(x)definite integral equation
- definite integral equation
input integrates(y(t)/sqrt(x-t),t,0,x) = 2y for
`int_0^x (y(t))/sqrt(x-t)` dt = 2ydifferential integral equation
- input ds(y)-ints(y) -y-exp(x)=0 for
`dy/dx-int y dx -y-exp(x)=0`fractional differential equation
dsolve( ) also solves fractional differential equation - Solve linear equations:
`d^0.5/dx^0.5 y = 2y`
`d^0.5/dx^0.5 y -y - E_(0.5) (4x^0.5) = 0`
`d^0.5/dx^0.5 y -y -exp(4x) = 0`
`(d^0.5y)/dx^0.5=sin(x)` - Solve nonlinear equations:
`d^0.5/dx^0.5 y = y^2`fractional integral equation
- `d^-0.5/dx^-0.5 y = 2y`
fractional differential integral equation
- ds(y,x,0.5)-ints(y,x,0.5) -y-exp(x)=0
`(d^0.5y)/(dx^0.5)-int y (dx)^0.5 -y-exp(x)=0`complex order differential equation
-
`(d^(1-i) y)/dx^(1-i)-2y-exp(x)=0`
variable order differential equation
-
`(d^sin(x) y)/dx^sin(x)-2y-exp(x)=0`
system of differential equations
- system of 2 equations with 2 unknowns x and y with a variable t :
dsolve( ds(x,t)=x-2y,ds(y,t)=2x-y ) - nonlinear equations:
dsolve( dx/dt=x-2y^2,dy/dt=2x^2-y ) - the 2 order system of 2 equations with 2 unknowns x and y with a variable t :
dsolve( x(2,t)=x,y(2,t)=2x-y ) - the 0.5 order system of 2 equations with 2 unknowns x and y with a variable t :
dsolve( x(0.5,t)=x,y(0.5,t)=x-y )partial differental equation
Math handbook chapter 14 partial differential equation.
PDE( ) and pdsolve( ) solve partial differental equation with two variables t and x, then click the plot2D button to plot solution, pull the t slider to change the t value. click the plot3D button for 3D graph. - Solve a linear equation:
`dy/dt = dy/dx-2y` - Solve a nonlinear equation:
`dy/dt = dy/dx*y^2`
`dy/dt = dy/dx-y^2`
`(d^2y)/(dt^2) = 2* (d^2y)/(dx^2)-y^2-2x*y-x^2`fractional partial differental equation
PDE( ) and pdsolve( ) solve fractional partial differental equation. - Solve linear equations:
`(d^0.5y)/dt^0.5 = dy/dx-2y` - Solve nonlinear equations:
`(d^0.5y)/dt^0.5 = 2* (d^0.5y)/dx^0.5*y^2`
`(d^0.5y)/dt^0.5 = 2* (d^0.5y)/dx^0.5-y^2`
`(d^1.5y)/(dt^1.5) + (d^1.5y)/(dx^1.5)-2y^2-4x*y-2x^2 =0`More examples are in Analytical Solution of Fractional Differential Equations
test solution
test solution for algebaic equation
test solution for algebaic equation to the unknown x by test( solution,eq, x) or click the test button : - test(1,x^2-1=0,x)
test( -1, x^2-5*x-6 )test solution for differential equation
test solution for differential equation to the unknown y by test( solution, eq ) or click the test button : - test( exp(2x), `dy/dx=2y` )
- test( exp(4x), `(d^0.5y)/dx^0.5=2y` )
test solution for recurrence equation to the unknown y
by rtest( solution, eq ) or click the rtest button.test solution for recurrence equation to the unknown f
by ftest( solution, eq ) or click the ftest button. - First graph is in real domain, second graph is in Laplace domain by Lapalce transform
laplace(x) - Input your function, click the laplace button :
laplace(sin(x))inverse laplace transform
- First graph is in Laplace domain , second graph is in real domain by inverse Lapalce transform
inverselaplace(1/x^2)Fourier transform
- First graph is in real domain, second graph is in Fourier domain by Fourier transform
fourier(x)Input your function, click the Fourier button :
- fourier(exp(x))
- sine wave
- Weierstrass function animation
convolution transform
First graph is in real domain, second graph is in convolution domain by convolution transform convolute(x) with x by default: -
Input your function, click the convolute button :
convolute(exp(x)) - convoute exp(x) with 1/sqrt(x) :
convolute(exp(x),1/sqrt(x)) - Input harmonic(2,x), click the defintion( ) button to show its defintion, check its result by clicking the simplify( ) button,
then click the limoo( ) button for its limit as x->oo.
Difference
- Δ(k^2) = difference(k^2)
Check its result by the sum( ) buttonSummation ∑
Indefinite sum
- ∑ k = sum(k)
- Check its result by the difference( ) button
Δ sum(k) = difference( sum(k) ) - In order to auto plot, the index variable should be x :
`sum_x x` = sum(x,x)definite sum
- Definite sum = Partial sum x from 1 to x :
1+2+ .. +x = `sum _(k=1) ^x k` = sum(k,k,1,x) - Definite sum, sum x from 1 to 5 :
1+2+ .. +5 = ∑(x,x,1,5) = sum(x,x,1,5)
sum(x^k,k,1,5)Definite sum with parameter x as upper limit
sum(k^2, k,1, x) - Check its result by the difference( ) button, and then the expand( ) button.
- convert to sum series definition :
tosum( exp(x) ) - expand above sum series by the expand( ) button
:
expand( tosum(exp(x)) )
Indefinite sum
- ∑ k
- sum( x^k/k!,k )
-
partial sum of 1+2+ .. + k
= ∑ k
= partialsum(k)
-
Definite sum of 1+2+ .. +5 = ∑ k
partial sum with parameter upper limit x
- sum(1/k^2,k,1,x)
infinite sum
- sum from 1 to oo:
Infinite sum of 1/1^2+1/2^2+1/3^2 .. +1/k^2+... = sum( 1/k^2,k,1,oo ) - sum from 0 to oo:
Infinite sum of 1/0!+1/1!+1/2! .. +1/k!+... = sum( 1/k!,k,0,oo ) - Infinite sum x from 0 to inf :
1/0!+1/1!+1/2!+ .. +1/x! = sum 1/(x!) as x->oo - convert to sum series definition :
tosum( exp(x) ) = toseries( exp(x) ) - check its result by clicking the simplify( ) button :
simplify( tosum( exp(x) )) - expand above sum series :
expand( tosum(exp(x)) ) - compare to Taylor series :
taylor( exp(x), x=0, 8) - compare to series :
series( exp(x) ) - Taylor series expansion as x=0, by default x=0.
taylor( exp(x) as x=0 ) = taylor(exp(x)) - series expand not only to taylor series,
series( exp(x) )
but aslo to other series expansion,
series( zeta(2,x) )Product ∏
- prod(x,x)
- definition of function :
definition( exp(x) ) - check its result by clicking the simplify( ) button :
simplify( def(exp(x)) ) - convert to series definition :
toseries( exp(x) ) - check its result by clicking the simplify( ) button :
simplify( tosum(exp(x)) ) - convert to integral definition :
toint( exp(x) ) - check its result by clicking the simplify( ) button :
simplify( toint(exp(x)) ) - numeric computation end with the equal sign =
sin(pi/4)= - numeric computation with the n( ) ≈≈ button:
n( sin(30 degree) )
n sin(30 degree) - JavaScript numeric calculator with the ≈ button
can calculate numeric, number theory, Probability, Statistics, matrix, solve equation.
more example in JavaScript mathjs
- numeric solve equation:
nsolve( x^2-5*x+6=0 )
nsolve( x^2-5*x+6 ) - numeric integrate, by default x from 0 to 1:
nint( x^2-5*x+6,x,0,1 )
nint x^2-5*x+6 as x from 0 to 1
nint sin(x) - numeric computation with the funplot ≈ button:
integrate(x=>sin(x),[1,2])more calculus operation in JavaScript calculus
- poly number:
poly(3,2) - Hermite number:
hermite(3,2) - harmonic number:
harmonic(-3,2)
harmonic(-3,2,4)
harmonic(1,1,4) = harmonic(1,4) = harmonic(4) - Bell number:
n(bell(5)) - double factorial 6!!
- Calculate the 4nd prime prime(4)
- is prime number? isprime(12321)
- next prime greater than 4 nextprime(4)
- binomial number `((4),(2))`
- combination number `C_2^4`
- harmonic number `H_4`
- congruence equation:
3x-1 = 2*(mod 2)
x^2-3x-2 = 2mod( 2) - modular equation:
Enter mod(x-1,2)=1 for
(x-1) mod 2 = 1
Enter mod(x^2-5x+7,2)=1 for
(x^2-5x+7) mod 2 = 1
Enter mod(x^2-5x+6,2)=0 for
(x^2-5x+6) mod 2 = 0 - Diophantine equation:
number of equation is less than number of the unknown, e.g. one equation with 2 unkowns x and y,
solve( 3x-2y-2=0, x,y )
solve( x^2-3x-2y-2=0, x,y )more example in JavaScript mathjs
- probability of standard normal distribution P( )
:
P(x<0.8) - standard normal distribution function Phi(x)
:
`Phi(x)` - solve Probability equation for k :
solve(P(x>k)=0.2,k) - combination(4,x)
combination(4,2)more example in JavaScript mathjs
- differentiate :
d(x and x*x) - integrate :
int(x and x*x)list
There are 2 types of list: [1,2,3] is numeric list in JavaScript, and the list(1,2,3) function is symbolic list in mathHand. The symbolic list element can be symbol, formula and function. We can sort list with - mathHand calculator on the = button :
sort(list(2,3,1)) - JavaScript calculator on the ≈ button :
[1,2,3].sort()more example in JavaScript mathjs
Statistics 统计
math handbook chapter 16 Statistics - sort(list( )), add numbers together by total(list()), max(list()), min(list()), size(list()) with mathHand calculator on the = button. e.g.
total(list(1,2,3)) - with JavaScript numeric calculator on the ≈ button :
sum([1,2])more example in JavaScript mathjs
Linear Algebra 线性代数 >>
vector
math handbook chapter 8 vectorIt has direction. the position of the element is fix so we cannot sort it. numeric vector is number with direction. the system auto plot the 2-dimentional vector. two vector( ) in the same dimention can be operated by +, -, *, /, and ^, the result can be checked by its reverse operation.
vector equation
- solve vector(1,2)+x=vector(2,4) is as same as x=vector(2,4)-vector(1,2)
- solve 2x-vector(2,4)=0 is as same as x=vector(2,4)/2
- solve 2/x-vector(2,4)=0 is as same as x=2/vector(2,4)
- solve vector(1,2)*x-vector(2,4)=0 is as same as x=vector(2,4)/vector(1,2)
- solve vector(1,2)*x-20=0 is as same as x=20/vector(1,2)
- solve vector(2,3)*x+vector(3,2)*y=vector(1,1),x,y
is as same as
solve(-1+2*x+3*y=0,-1+3*x+2*y=0)
vector calculus
- differentiate vector(x,x) :
d(vector(x,x)) - differentiate sin(vector(x,x)) :
d(sin(vector(x,x)))Matrix 复数矩阵
math handbook chapter 4 matrixComplex matrix [[1,2],[3,4]] can be operated by +,-,*,/,^,
- with matrix calculator 复数矩阵计算器
- with JavaScript numeric calculator ≈
more example in JavaScript mathjs
programming 编程 >>
There are many coding : - math coding 数学编程
- HTML + JavaScript coding 网页编程
- cloud computing = web address coding 云计算 = 网址编程 = 网址计算器
Graphics >>
- graphics
- Classification by plot function 按制图函数分类
- Classification by appliaction 按应用分类
Plot 制图 >>
- plane curve 2D
- surface 2D
3D graph 立体图 plot 3D >>
- space curve 3D
- surface 3D
- surface 4D
Drawing 画画 >>
- drawing
Arithmetic 算术 >>
Exact computation
Convert to complex( )
more are in numeric math
Algebra 代数 >>
tangent
inverse function
Function 函数 >>
Trigonometry 三角函数
Complex Function 复变函数
math handbook chapter 10 Complex Function 复变函数complex2D( ) shows the real and imag curves in real domain x, and complex3D( ) shows complex function in complex domain z, for 20 graphes in one plot.
Calculus 微积分 >>
Limit
math handbook chapter 4.1 limit
differentiate graphically
some functions cannot be differentiated or integrated symbolically, but can be semi-differentiated and integrated graphically in diff2D.Integrals
Math Handbook chapter 6 integral calculusEquation 方程 >>
inverse an equation
bugs >>
There are over 600 bugs in wolfram software but they are no problem in MathHand.comTransform >>
Math handbook chapter 11 integral transformlaplace transform
Discrete Math 离散数学 >>
Series 级数
`prod x`
Definition 定义式 >>
series definition
integral definition
Numeric math 数值数学 >>
Number Theory 数论 >>
When the variable x of polynomial is numnber, it becomes polynomial number :
Probability 概率 >>