Green's theorem — Safekipedia Discoverer

Green's theorem is an important idea in mathematics, especially in a part called vector calculus. It helps us connect two types of integrals: line integrals and double integrals. A line integral looks at something along a curve, like the edge of a shape, while a double integral looks at something spread out over an area.

This theorem tells us that we can often change a tricky line integral around a closed curve into a simpler double integral over the area inside that curve. This can make solving problems much easier!

Green's theorem is a special case of a bigger idea called Stokes' theorem, which works in three dimensions. It's also closely related to the divergence theorem and the fundamental theorem of calculus. The theorem is named after George Green, a mathematical physicist who did important work in this area.

Theorem

Green's theorem helps us connect two types of integrals. It says that a line integral around a closed curve in a plane is the same as a double integral over the area inside that curve. This theorem is useful in vector calculus and connects ideas from two dimensions to more general theories in higher dimensions.

Application

Green's theorem helps us understand how things flow around and through shapes in a flat plane. It connects two types of calculations: one that follows the edges of a shape (called a line integral) and another that covers the area inside the shape (called a double integral).

The theorem has two main uses. The first, called the circulation form, helps measure how something spins or moves around the edge of a shape. The second, called the flux form, helps measure how much of something flows out through the edge of the shape. These ideas are useful in studying movement and flow in two dimensions.

Proof when D is a simple region

Green's theorem is a way to connect two kinds of mathematical sums: one that follows a path (called a line integral) and another that covers an area (called a double integral). This theorem helps us understand how these two types of sums are related.

For a simple shape called a "type I region," we can prove part of Green's theorem by looking at a special kind of area. We break the path around this area into four smaller paths and study each one. By adding up the results from these paths, we can show that they match the result of the area sum. This same idea works for other shapes too, helping us prove Green's theorem for more complex regions.

∮ C L d x = ∬ D ( − ∂ L ∂ y ) d A {\displaystyle \oint _{C}L\,dx=\iint _{D}\left(-{\frac {\partial L}{\partial y}}\right)dA}

∮ C M d y = ∬ D ( ∂ M ∂ x ) d A {\displaystyle \oint _{C}\ M\,dy=\iint _{D}\left({\frac {\partial M}{\partial x}}\right)dA}

∬ D ∂ L ∂ y d A = ∫ a b ∫ g 1 ( x ) g 2 ( x ) ∂ L ∂ y ( x , y ) d y d x = ∫ a b [ L ( x , g 2 ( x ) ) − L ( x , g 1 ( x ) ) ] d x . {\displaystyle {\begin{aligned}\iint _{D}{\frac {\partial L}{\partial y}}\,dA&=\int _{a}^{b}\,\int _{g_{1}(x)}^{g_{2}(x)}{\frac {\partial L}{\partial y}}(x,y)\,dy\,dx\\&=\int _{a}^{b}\left[L(x,g_{2}(x))-L(x,g_{1}(x))\right]\,dx.\end{aligned}}}

∮ C L d x = ∫ C 1 L ( x , y ) d x + ∫ C 2 L ( x , y ) d x + ∫ C 3 L ( x , y ) d x + ∫ C 4 L ( x , y ) d x = ∫ a b L ( x , g 1 ( x ) ) d x − ∫ a b L ( x , g 2 ( x ) ) d x . {\displaystyle {\begin{aligned}\oint _{C}L\,dx&=\int _{C_{1}}L(x,y)\,dx+\int _{C_{2}}L(x,y)\,dx+\int _{C_{3}}L(x,y)\,dx+\int _{C_{4}}L(x,y)\,dx\\&=\int _{a}^{b}L(x,g_{1}(x))\,dx-\int _{a}^{b}L(x,g_{2}(x))\,dx.\end{aligned}}}

Proof for rectifiable Jordan curves

Green's theorem is a special case of Stokes' theorem that helps us understand how integrals along a closed curve relate to integrals over the area it encloses. It says that for certain well-behaved curves and functions, the total along the curve can be found by looking at the area inside instead.

The theorem works for curves that are smooth enough and enclose a region in the plane. It connects line integrals around the edge to double integrals over the area inside, making calculations easier in many situations. This idea is important in physics and engineering for understanding flow and other quantities around boundaries.

Jordan curve R

Validity under different hypotheses

Green's theorem can work under different conditions. One common set of conditions requires that two functions, called A and B, are continuous and have certain types of derivatives at every point. This means we can measure how these functions change in different directions.

As a result of these conditions, we get a special result called the Cauchy Integral Theorem. This theorem says that if we have a special kind of curve and a function that changes smoothly inside that curve, then the integral of that function around the curve will be zero. This helps us understand how functions behave in complex planes.

Multiply-connected regions

Green's theorem can also be used for more complex shapes that have "holes" or spaces inside them. Imagine a shape with one big outer edge and several smaller inner edges that form holes. The theorem still works, but we need to subtract the line integrals around each of these inner edges from the outer edge.

The result tells us that the total of these adjusted line integrals around the outer and inner edges equals a double integral over the area between them, just like with simpler shapes. This helps us understand how integrals behave around more complicated, multiply-connected regions.

Relationship to Stokes' theorem

Green's theorem is a special version of the Kelvin–Stokes theorem when we look at a flat area in a two-dimensional space. We can think of it like turning a flat shape into a very thin three-dimensional object where everything stays flat — there’s no height or depth.

This helps us connect two different kinds of math ideas: one that looks at paths around the shape’s edge, and another that looks at the area inside the shape. In simpler terms, Green’s theorem shows how we can understand something happening around the outside of a shape by looking at what’s happening inside it.

Relationship to the divergence theorem

Green's theorem is closely related to the divergence theorem in two dimensions. It connects a line integral around a closed curve to a double integral over the area enclosed by that curve. This relationship helps us understand how certain properties of vector fields behave in two-dimensional space.

The theorem shows that calculations involving curves can sometimes be easier done by looking at the area they surround instead. This idea is useful in many areas of physics and engineering where such integrals appear.

Area calculation

Green's theorem helps us find the area of a shape by using a line around it. Imagine tracing the edge of a flat region with a pencil; the theorem lets us calculate the area inside by looking at this path.

One way to do this is by choosing two special values, L and M, that follow a particular rule. With these, the area can be found using a loop around the shape. There are also simple formulas that use the positions along the edge to compute the area.

History

William Thomson, also known as Lord Kelvin, named this theorem after George Green. Green first shared a similar idea in an 1828 paper about electricity and magnetism, but it was almost forgotten until Thomson found it again in 1845. Later, in 1846, Augustin-Louis Cauchy published the theorem in one of his papers. Finally, Bernhard Riemann provided a proof of the theorem in his doctoral dissertation in 1851.