MATH 251 Fall 2023 Section 16 5 Curl and Divergence Interpretation of curl and divergence Let F x y z P x y z i Q x y z j R x y z k be the velocity of a uid at the point x y z 1 curl F x y z is a three dimensional vector whose magnitude measures how fast the uid rotates about the axis that points in the direction of curl F x y z 2 div F measures the tendency of the uid to diverge from the point x0 y0 z0 In particular div F 0 means more uid leaves x0 y0 z0 than enters x0 y0 z0 div F 0 means more uid enters x0 y0 z0 than leaves x0 y0 z0 div F 0 means the same amount of uid enters x0 y0 z0 as it leaves x0 y0 z0 Figure 1 The left most has div F 0 the middle has div F 0 the right most has div F 0 1 2 Curl Given the vector eld F P i Qj Rk then the curl of F is curl F R y Q z i P z R x j Q x An easy way to remember the above formula is curl F r F Example 1 If F x y z x sin z i y cos x j z2k nd curl F P y k j y k i x z P Q R Theorem 1 If f is a function of three variables that has continuous second order partial derivatives then As a result if a vector eld F is conservative then curl rf 0 curl F 0 Theorem 1 gives us a way of verifying that a vector eld is not conservative Example 2 Show that the vector eld F x y z x sin z i y cos x j z2k is not conservative 3 The converse of Theorem 1 is not true in general that is even when curl F 0 F may NOT be conservative For example the vector eld F y x2 y2 0 has curl F 0 but F is not conservative becauseRC F dr 2 where C is the closed curve r t h cos t sin t 0i for 0 t 2 If F were conservative then the above integral would be 0 x2 y2 x However the converse of Theorem 1 is true if the domain of F contains no hole Theorem 2 If F is a vector eld de ned on all of R3 whose component functions have continuous partial derivatives and curl F 0 then F is conservative Example 3 Show that is conservative Find its potential function F x y z 2xy yz i x2 xz j xyk 4 Divergence If F P i Qj Rk then the divergence of F is An easy way to remember the formula is div F P x Q y R z div F r F Example 4 Find the divergence of F x y z hxy2 ey sin x yz2i Find and interpret the divergence at the point 2 1 1 Theorem 3 If F P i Qj Rk and P Q R have continuous second order partial derivatives then div curl F 0 Example 5 Show that F x y z hxy2 2zx3 ezi is not the curl of another vector eld
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