Định lý Green

Trong toán học, định lý Green' đưa ra mối liên hệ giữa tích phân đường quanh một đường cong khép kín C vàa tích phân mặt trên một miền D bao quanh bởi C. Đây là trường hợp đặt biệt trong không gian 2 chiều của định lý Stokes, và được đặt tên theo nhà toán học người Anh tên George Green.

Định lý

C là một đường đơn đóng có định hướng dương trong mặt phẳng $\mathbb {R}$², và D là miền được bao quanh bởi C. Nếu L và M là các hàm số với biến (x, y) được định nghĩa trên miền mở chứa D và có các đạo hàm riêng phần liên tục trên đó, thì

$\oint _{C}(L\,\mathrm {d} x+M\,\mathrm {d} y)=\iint _{D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\,\mathrm {d} x\,\mathrm {d} y.$

Relationship to the Stokes theorem

Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the xy-plane:

We can augment the two-dimensional field into a three-dimensional field with a z component that is always 0. Write F for the vector-valued function $\mathbf {F} =(L,M,0)$. Start with the left side of Green's theorem:

$\oint _{C}(L\,dx+M\,dy)=\oint _{C}(L,M,0)\cdot (dx,dy,dz)=\oint _{C}\mathbf {F} \cdot d\mathbf {r} .$

Then by Kelvin–Stokes Theorem:

$\oint _{C}\mathbf {F} \cdot d\mathbf {r} =\iint _{S}\nabla \times \mathbf {F} \cdot \mathbf {\hat {n}} \,dS.$

The surface $S$ is just the region in the plane $D$, with the unit normals $\mathbf {\hat {n}}$ pointing up (in the positive z direction) to match the "positive orientation" definitions for both theorems.

The expression inside the integral becomes

$\nabla \times \mathbf {F} \cdot \mathbf {\hat {n}} =\left[\left({\frac {\partial 0}{\partial y}}-{\frac {\partial M}{\partial z}}\right)\mathbf {i} +\left({\frac {\partial L}{\partial z}}-{\frac {\partial 0}{\partial x}}\right)\mathbf {j} +\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\mathbf {k} \right]\cdot \mathbf {k} =\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right).$

Thus we get the right side of Green's theorem

$\iint _{S}\nabla \times \mathbf {F} \cdot \mathbf {\hat {n}} \,dS=\iint _{D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\,dA.$

Relationship to the divergence theorem

Considering only two-dimensional vector fields, Green's theorem is equivalent to the following two-dimensional version of the divergence theorem:

$\iint _{D}\left(\nabla \cdot \mathbf {F} \right)dA=\oint _{C}\mathbf {F} \cdot \mathbf {\hat {n}} \,ds,$

where $\mathbf {\hat {n}}$ is the outward-pointing unit normal vector on the boundary.

To see this, consider the unit normal $\mathbf {\hat {n}}$ in the right side of the equation. Since in Green's theorem $d\mathbf {r} =(dx,dy)$ is a vector pointing tangential along the curve, and the curve C is the positively-oriented (i.e. counterclockwise) curve along the boundary, an outward normal would be a vector which points 90° to the right, which would be $(dy,-dx)$. The length of this vector is ${\sqrt {dx^{2}+dy^{2}}}=ds$. So $\mathbf {\hat {n}} \,ds=(dy,-dx).$

Now let the components of $\mathbf {F} =(P,Q)$. Then the right hand side becomes

$\oint _{C}\mathbf {F} \cdot \mathbf {\hat {n}} \,ds=\oint _{C}Pdy-Qdx$

which by Green's theorem becomes

$\oint _{C}-Qdx+Pdy=\iint _{D}\left({\frac {\partial P}{\partial x}}+{\frac {\partial Q}{\partial y}}\right)\,dA=\iint _{D}\left(\nabla \cdot \mathbf {F} \right)dA.$

The converse can also easily shown to be true.

Area Calculation

Green's theorem can be used to compute area by line integral. The area of D is given by:

$A=\iint _{D}\mathrm {d} A.$

Provided we choose L and M such that:

${\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}=1.$

Then the area is given by:

$A=\oint _{C}(L\,\mathrm {d} x+M\,\mathrm {d} y).$

Possible formulas for the area of D include:

$A=\oint _{C}x\,\mathrm {d} y=-\oint _{C}y\,\mathrm {d} x={\tfrac {1}{2}}\oint _{C}(-y\,\mathrm {d} x+x\,\mathrm {d} y).$

See also

• Planimeter
• Method of image charges – A method used in electrostatics that takes advantage of the uniqueness theorem (derived from Green's theorem)

Further reading

• Calculus (5th edition), F. Ayres, E. Mendelson, Schuam's Outline Series, 2009, ISBN 978-0-07-150861-2.
• Advanced Calculus (3rd edition), R. Wrede, M.R. Spiegel, Schuam's Outline Series, 2010, ISBN 978-0-07-162366-7.

External links

• Green's Theorem on MathWorld