# 4: Integration in Vector Fields

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• 4.1: Differentiation and Integration of Vector Valued Functions
All of the properties of differentiation still hold for vector values functions. Moreover because there are a variety of ways of defining multiplication, there is an abundance of product rules.
• 4.2: Surfaces and Area
In first year calculus we have seen how to find the surface area of revolution. Now that we haw the power of double integration, we are ready to take on the surface area for more general surfaces.
• 4.3: Line Integrals
This section covers the integration of a line over a 3-D scalar field. A line integral takes two dimensions, combines it the sum of all the arc lengths that the line makes, and then integrates the functions of x and y over that constructed line.
• 4.4: Conservative Vector Fields and Independence of Path
Conservative Vector Fields have unique and powerful aspect that can simplify calculations.
• 4.5: Path Independence, Conservative Fields, and Potential Functions
For certain vector fields, the amount of work required to move a particle from one point to another is dependent only on its initial and final positions, not on the path it takes. Gravitational and electric fields are examples of such vector fields. This section will discuss the properties of these vector fields.
• 4.6: Vector Fields and Line Integrals: Work, Circulation, and Flux
This section demonstrates the practical application of the line integral in Work, Circulation, and Flux.
• 4.7: Surface Integrals
Up until this point we have been integrating over one dimensional lines, two dimensional domains, and finding the volume of three dimensional objects. In this section we will be integrating over surfaces, or two dimensional shapes sitting in a three dimensional world. These integrals can be applied to real world situations such as finding the upward force on an open parachute.
• 4.8: Green’s Theorem in the Plane
Green's Theorem, allows us to convert the line integral into a double integral over the region enclosed by C
• 4.9: The Divergence Theorem and a Unified Theory
When we looked at Green's Theorem, we saw that there was a relationship between a region and the curve that encloses it. This gave us the relationship between the line integral and the double integral. Now consider the following theorem:
• 4.10: Stokes’ Theorem
In this section we see the generalization of a familiar theorem, Green’s Theorem. Just as before we are interested in an equality that allows us to go between the integral on a closed curve to the double integral of a surface. Some important definitions to know before proceeding are: simple closed curve, divergence, flux, curl, and normal vector. Knowing how to calculate the determinant of 2x2 and 3x3 matrices will also help deepen your understanding of divergence and curl.

4: Integration in Vector Fields is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.