Question

Complete the proof of the special case of Green's Theorem by proving Equation 3.

Short answer

The special case of Green's theorem is proved.

Step-by-step solution

Step 1:Green's theorem

Consider the expression for special case of Green's theorem as,

\(\int_{C}{Q}dy=\iint_{D}{\frac{\partial Q}{\partial x}}dA\)

\({f_1}\) and \({f_2}\) are continuous

Consider a type II region as \(D = \left\{ {(x,y)\mid {f_1}(y) \le x \le {f_2}(y),c \le y \le d} \right\}\) and functions \({f_1}\) and \({f_2}\) are continuous. The type II region is shown in Figure \(1.\)

curves are

From Figure 1, the curves are,

\(\begin{aligned}{l}\int_{{C_1}} Q dy & = \int_d^c Q \left( {{f_1}(y),y} \right)dy\\\int_{{C_2}} Q dy & = 0\\\int_{{C_3}} Q dy & = \int_c^d Q \left( {{f_2}(y),y} \right)dy\\\int_{{C_4}} Q dy & = 0\end{aligned}\)

Find the value of and\(\)\(\int_C Q dy\).

Find the value of .\(\int_{C}{Q}dy\)

\(\begin{align} & \int_{C}{Q}dy=\int_{{{C}_{1}}+{{C}_{2}}+{{C}_{3}}+{{C}_{4}}}{Q}dy \\ & =\int_{{{C}_{1}}}{Q}dy+\int_{{{C}_{2}}}{Q}dy+\int_{{{C}_{3}}}{Q}dy+\int_{{{C}_{4}}}{Q}dy \end{align}\)

\(\)Find the value of \(\int_C Q dy\).

\(\begin{aligned}\int_C Q dy & = \int_{{C_1} + {C_2} + {C_3} + {C_4}} Q dy\\ & = \int_{{C_1}} Q dy + \int_{{C_2}} Q dy + \int_{{C_3}} Q dy + \int_{{C_4}} Q dy\end{aligned}\)

substitute

Equate the equations (2) and (1). \(\int_C Q dy = \mathop \int\!\!\!\int \nolimits_D^{\partial Q} \frac{{\partial x}}{{\partial x}}dA\)

Hence, the special case of Green's theorem is proved.