- The integral of a function
denotes the area under its curve. (If it is below zero, it is negative). It is more formally defined as Where the set ofis a set of points spaced evenly. The intuition is that we take thin slices of the function. - By the Fundamental Theorem of Calculus, the derivative and the integral are inverses of each other.
- Iterated integrals generalize integrals in that they take the volume under the surface of the function. The intuition is to take thin slices along each direction.
- The line integral is an integral over a Vector Field
along a path . It is denoted as 1 The RHS of the above is a parameterization over, a function denoting the position at time if we perform a motion on . -
The Fundamental Theorem of Line Integrals says that for a curve
starting at point and ending at point , we have that That is, gradient fields are path independent, that is, the above integral holds for any
that starts and ends at and . This also implies the closed loop integral is defined as
That is, gradient fields are conservative.
This also implies that
. -
Green’s Theorem says that if
is a closed curve oriented counter clockwise enclosing region and a two-dimensional vector field defined and differentiable in . Then -
Green’s Theorem for flux says that if
is a closed curve oriented counter clockwise enclosing region and a two-dimensional vector field defined and differentiable in . Then
-
-
The surface integral is an extension of the line integral, wherein we integrate over a surface
. denotes the surface area element. -
For a surface
, the flux measures the amount of the function that passes through the surface (if we’re dealing with lines, we simply use line integrals). It is defined as Where
is a unit normal vector pointing perpendicular to the surface -
The Divergence Theorem states that that for closed surface
enclosing region and a vector field defined and differentiable everywhere in , then Where
is oriented outwards from the surface . Informally, the sum of all sources of the field in a region gives the net flux in the region
-
-
Stokes’ Theorem is a generalization of Green’s Theorem for three dimensions. It states that for any line
in space enclosing surface , we have Informally, the line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface.
Note: the surface must be oriented following the right hand rule (thumb towards the curve, index towards the interior of the surface, middle finger points to normal).
The choice of surface doesn’t matter as long as
bounds it.