Section 5
Vector Calculus
By Boundless
![Thumbnail](../../../../../../figures.boundless-cdn.com/17866/raw/vectorfield.jpg)
A vector field is an assignment of a vector to each point in a subset of Euclidean space.
![Thumbnail](../../../../../../figures.boundless-cdn.com/17872/square/e-integral-of-scalar-field.gif)
A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
![Thumbnail](../../../../../../figures.boundless-cdn.com/23733/square/e-integral-of-scalar-field.gif)
A line integral is an integral where the function to be integrated is evaluated along a curve.
![Thumbnail](../../../../../../figures.boundless-cdn.com/17873/raw/ge-charge-plane-horizontal.jpg)
Gradient theorem says that a line integral through a gradient field can be evaluated from the field values at the endpoints of the curve.
![Thumbnail](../../../../../../figures.boundless-cdn.com/18401/raw/-27s-theorem-simple-region.jpg)
Green's theorem gives relationship between a line integral around closed curve
![Thumbnail](../../../../../../figures.boundless-cdn.com/17875/square/fig1.jpeg)
The four most important differential operators are gradient, curl, divergence, and Laplacian.
![Thumbnail](../../../../../../figures.boundless-cdn.com/17878/raw/stokes-27-theorem.jpg)
A parametric surface is a surface in the Euclidean space
![Thumbnail](../../../../../../figures.boundless-cdn.com/23735/raw/stokes-27-theorem.jpg)
The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field.
![Thumbnail](../../../../../../figures.boundless-cdn.com/23736/raw/stokes-27-theorem.jpg)
Stokes' theorem relates the integral of the curl of a vector field over a surface to the line integral of the field around the boundary.
![Thumbnail](../../../../../../figures.boundless-cdn.com/17879/raw/aceswithandwithoutboundary.jpg)
The divergence theorem relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.