Solve the Problem $F$ Of a Fluid Is the Spin Field $F = - \frac { k y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \mathrm { i } + \frac { k x } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \mathrm { j }$

question 47

Essay

Solve the problem.
-The velocity field $F$ of a fluid is the spin field $F = - \frac { k y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \mathrm { i } + \frac { k x } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \mathrm { j }$ . Following the smooth curve $y = f ( x )$ from $( a , f ( a ) )$ to $( b , f ( b ) )$ , show that the flux across the curve is $\int _ { C } F \cdot n d s = k \left[ \left( a ^ { 2 } + ( f ( a ) ) ^ { 2 } \right) ^ { 1 / 2 } - \left( b ^ { 2 } + ( f ( b ) ) ^ { 2 } \right) ^ { 1 / 2 } \right]$

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