Mathematics
Calculus: Curl ∇⨯ & Div ∇⋅
Curl ∇⨯ and Div ∇⋅ in three-dimensional Cartesian coordinates
For a given vector function F : ℝ3 → ℝ3 , F = (Fx , Fy , Fz ) (x,y,z), describing the vector field in 3-dimensional Cartesian space, curl (rotor) is a
differentiation vector operator, determining the circulation density at each point of the field. It is defined as
curl (F) = rot (F) = ∇⨯F =
= ( - , - , - )
Divergence is a differentiation vector operator that produces a scalar field describing the rate that the vector field F changes the volume in an
infinitesimal neighborhood of each point. It is defined as
div (F) = ∇⋅F = ( , , ) ( Fx , Fy , Fz ) = + +
Important Note:
Please be very careful to write a multiplication symbol '*' when writing a term consisting of multiplication of two (or more) variables x*y or parameters a*b.
Terms xy or ab are instances of a single variable 'xy' or 'ab' whose label consists of two characters. See Syntax Help for a proper syntax for writing a formula.
Please enter the vector function F ( , , ) you want to differentiate:For a given vector function F : ℝ3 → ℝ3 , F = (Fx , Fy , Fz ) (x,y,z), describing the vector field in 3-dimensional Cartesian space, curl (rotor) is a
differentiation vector operator, determining the circulation density at each point of the field. It is defined as
curl (F) = rot (F) = ∇⨯F =
| î | ĵ | k̂ |
| ∂/∂x | ∂/∂y | ∂/∂z |
| Fx | Fy | Fz |
∂Fz
∂y
∂Fy
∂z
∂Fx
∂z
∂Fz
∂x
∂Fy
∂x
∂Fx
∂y
Divergence is a differentiation vector operator that produces a scalar field describing the rate that the vector field F changes the volume in an
infinitesimal neighborhood of each point. It is defined as
div (F) = ∇⋅F = (
∂
∂x
∂
∂y
∂
∂z
∂Fx
∂x
∂Fy
∂y
∂Fz
∂z
Important Note:
Please be very careful to write a multiplication symbol '*' when writing a term consisting of multiplication of two (or more) variables x*y or parameters a*b.
Terms xy or ab are instances of a single variable 'xy' or 'ab' whose label consists of two characters. See Syntax Help for a proper syntax for writing a formula.
| Fx | = | |
| Fy | = | |
| Fz | = |
∇⨯F = curl(F) = ( curl x (F) , curl y (F) , curl z (F) ) :
| curl x (F) (x,y,z) | = | ||
| curl y (F) (x,y,z) | = | ||
| curl z (F) (x,y,z) | = | ||
| ∇⋅F = div (F) (x,y,z) | = | ||
Specify the point ( x , y , z ) for which you want to calculate curl (F) (x,y,z) and div (F) (x,y,z)( x , y , z ) = ( , , )
| curl (F) (x,y,z) | = | ( , , ) |
| div (F) (x,y,z) | = |
Curl ∇⨯ and Div ∇⋅ in two-dimensional Cartesian coordinates
For a given vector function F : ℝ2 → ℝ2 , F = (Fx , Fy) (x,y), describing the vector field in 2-dimensional Cartesian space, curl (rotor) is a
differentiation vector operator, determining the circulation density at each point of the field. It is defined as
curl (F) = rot (F) = ∇⨯F =
= -
Divergence is a differentiation vector operator that produces a scalar field describing the rate that the vector field F changes the volume in an
infinitesimal neighborhood of each point. It is defined as
div (F) = ∇⋅F = ( , ) ( Fx , Fy ) = +
Important Note:
Please be very careful to write a multiplication symbol '*' when writing a term consisting of multiplication of two (or more) variables x*y or parameters a*b.
Terms xy or ab are instances of a single variable 'xy' or 'ab' whose label consists of two characters. See Syntax Help for a proper syntax for writing a formula.
Please enter the vector function F ( , ) you want to differentiate:For a given vector function F : ℝ2 → ℝ2 , F = (Fx , Fy) (x,y), describing the vector field in 2-dimensional Cartesian space, curl (rotor) is a
differentiation vector operator, determining the circulation density at each point of the field. It is defined as
curl (F) = rot (F) = ∇⨯F =
| ∂/∂x | ∂/∂y |
| Fx | Fy |
∂Fy
∂x
∂Fx
∂y
Divergence is a differentiation vector operator that produces a scalar field describing the rate that the vector field F changes the volume in an
infinitesimal neighborhood of each point. It is defined as
div (F) = ∇⋅F = (
∂
∂x
∂
∂y
∂Fx
∂x
∂Fy
∂y
Important Note:
Please be very careful to write a multiplication symbol '*' when writing a term consisting of multiplication of two (or more) variables x*y or parameters a*b.
Terms xy or ab are instances of a single variable 'xy' or 'ab' whose label consists of two characters. See Syntax Help for a proper syntax for writing a formula.
| Fx | = | |
| Fy | = |
| curl x (F) (x,y) | = | |
| ∇⋅F = div (F) (x,y) | = |
Specify the point ( x , y ) for which you want to calculate curl (F) (x,y) and div (F) (x,y)( x , y ) = ( , )
| curl (F) (x,y) | = | |
| div (F) (x,y) | = |
Formula Syntax:
Available Functions:| < formula > | := | <number> |<variable> |<operation> |<function>; |(<formula>) | |
| <differential formula> | ||
| < equation > | := | <formula>=<formula> |
| < differential equation > | := | <equation> |
| < operation > | := | <unary operation> |<binary operation> | |
| < unary operation > | := | -<formula> |
| < binary operation > | := | <formula><binary operator><formula> |
| < binary operator > | := | + |- |* |/ |^ |
| < differential formula > | := | <formula>' |d(<formula>) |∂(<formula>) | |
| < function > | := | <functor>(<arguments>) |
| < arguments > | := | <formula> |<formula>,<arguments> |
| < functor > | := | See list of available functions bellow |
| < number > | := | <unsigned number> |-<unsigned number> |
| < unsigned number > | := | <digits> | <digits>.<digits> | <special number> |
| < digits > | := | <digit> | <digit><digits> |
| < digit > | := | 0 | 1 |2 |3 |4 |5 |6 |7 |8 |9 |
| < special number > | := | pi | e |
| < variable > | := | <letter> | <letter><alphanumerics> |
| < alphanumerics > | := | <alphanumeric> | <alphanumeric><alphanumerics> |
| < alphanumeric > | := | <letter> | <digit> |
| < letter > | := | a | ... | z | A | ... | Z |
| Name | Description | # of arguments | |
| Trigonometric | |||
| sin | sine function of an angle | 1 | |
| cos | cosine function of an angle | 1 | |
| tan | tangent function of an angle | 1 | |
| cot | cotangent function of an angle | 1 | |
| sec | secant function of an angle | 1 | |
| csc | cosecant function of an angle | 1 | |
| Inverse Trigonometric | |||
| arcsin | inverse of sine function | 1 | |
| arccos | inverse of cosine function | 1 | |
| arctan | inverse of tangent function | 1 | |
| arccot | inverse of cotangent function | 1 | |
| arcsec | inverse of secant function | 1 | |
| arccsc | inverse of cosecant function | 1 | |
| Logaritmic | |||
| ln | natural logarithm to the base e | 1 | |
| log10 | logarithm to the base 10 | 2 | |
| log | logarithm, with 1st argument as a base, of the 2nd argument | 2 | |
| Power and Root | |||
| sqrt | square root | 1 | |
| root | first argument root of the second argument | 2 | |
| Miscellaneous | |||
| abs | absolute value | 1 | |
| max | maximum | 2 | |
| min | minimum | 2 | |
| sgn | sign or signum function | 1 | |
Please Note:
| |||
| Examples: | |||
| You must write: | Interpreted as: | ||
| tan(x) | tan(x) = sin(x)cos(x) | ||
| log(a,b) | loga(b) | ||
| ln(e^x) | ln ( e x ) = x | ||
| abs(-2) | | -2 | = 2 | ||
| cos(pi/3) | cos(π3) = 12 | ||
| root(x, y) | x √ y | ||
| y'=2*x*y | y' = 2xy | ||
External links: