91影视

The theorem (1), describing the line integral of a vector along a closed path, shows that for a closed path $\left(d\right)鈬�\left(a\right),\left(a\right)鈬�\left(c\right),\left(c\right)鈬�\left(b\right),\left(b\right)鈬�\left(c\right)$and $\left(c\right)鈬�\left(a\right)$.

The line integral of a vector F along a route dl is defined as$\mathbf{鈭�}\mathit{F}\mathbf{路}\mathit{d}\mathit{l}$ . If the line integral is independent of the path, then the line integral of function F must be zero along that closed path.

Let F be a gradient of a scalar function as $F=-鈭�v$ . Now find the curl of function F as,

$$\begin{gathered} 鈭�脳F=鈭�脳\left[-\left(鈭�v\right)\right] \\ =-鈭�脳\left[\left(鈭�v\right)\right] \\ =0 \end{gathered}$$

The result is zero because the curl of gradient of a vector is always zero. Thus we can say that $d鈬�a$.

According to strokes theorem,

$鈭�F路dl=鈭�\left(鈭�脳F\right)路da$

Since the curl of vector function F is given as zero, that is $鈭�脳F=0$. So, above equation becomes,

$$\begin{gathered} 鈭�F路dl=鈭�0路da \\ =0 \end{gathered}$$

Thus, we can say, $a鈬�c$.

The theorem (1) says that the path dl from a to b is an independent path. As discussed, $F=-鈭�v$, then using strokes theorem , we can write the following line integral,

${鈭�}_a^{b}F路dl=-{鈭�}_a^b\left(鈭�v\right)路dl$

Thus, the path from a to b is independent according to the gradient theorem (1). Hence, from$a鈬�b$.

As the line integral ${鈭�}_a^{b}F路dl$is independent of path from a to b , then for any closed loop, this line integral gives same value as the final and initial points in any closed loop coincide with each other. Thus, we can write, $b鈬�c$.

As discussed, if the line integral is path independent , then for any closed loop, the value of line integral $鈭�F路dl$is always zero, that is,role="math" localid="1657521793564" $鈭�F路dl=0$

According to strokes theorem,

$鈭�F路dl=鈭�\left(鈭�脳F\right)路da$

Since the line integral of vector function F is zero, that is, $鈭�F路dl=0$. So, above equation becomes,

$$\begin{gathered} 0=鈭�\left(鈭�脳F\right)路da \\ 鈭�脳F=0 \end{gathered}$$

Thus, we can say, $c鈬�a$.