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Section 17.5

Section 17.5. Curl and Divergence. THE DEL OPERATOR. The vector differential operator is defined as It has meaning when it operates on a scalar function to produce the gradient of f :. THE CURL.

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Section 17.5

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  1. Section 17.5 Curl and Divergence

  2. THE DEL OPERATOR The vector differential operator is defined as It has meaning when it operates on a scalar function to produce the gradient of f :

  3. THE CURL If F = P i + Q j + R k is a vector field on and the partial derivatives of P, Q, and R all exist, then the curl of F is the vector field on defined by

  4. CURL AND THE DEL OPERATOR

  5. THEOREM If f is a function of three variables that has continuous second-order partial derivatives, then NOTE: Since a conservative vector field is one for which , the above theorem can be rephrased as: If F is conservative, then curl F = 0.

  6. THEOREM Theorem: If F is a vector field defined on all of whose component functions have continuous partial derivatives and curl F = 0, then F is a conservative vector field. NOTE: More generally the theorem is true if the domain is simply-connected, that is, “has no hole.”

  7. AN INTERPRETATION OF CURL Suppose F represents the velocity field of a fluid flow. Particles near (x, y, z) in the fluid tend to rotate about the axis that points in the direction of curl F(x, y, z) and the length of this curl vector is a measure of how quickly the particles move around the axis. If curlF=0 at a point P, then the fluid is free from rotations at P and F is called irrotational at P. In other words there is no whirlpool or eddy at P.

  8. DIVERGENCE If F = P i + Q j + R k is a vector field on and the partial derivatives of P, Q, and R all exist, then the divergence ofF is the function of three variables defined by NOTE:

  9. THEOREM If F = P i + Qj + R k is a vector field on and P, Q, and R have continuous second-order partial derivatives, then

  10. AN INTERPRETATION OF DIVERGENCE If F(x, y, z) is the velocity of a fluid (or gas), then div F(x, y, z) represents the net rate of change (with respect to time) of the mass of fluid (or gas) flowing from the point (x, y, z) per unit volume. In other words, div F(x, y, z) measures the tendency of the fluid to diverge from the point (x, y, z). If div F > 0, the fluid diverges away from the point. If div F < 0, the fluid accumulates toward the point. If div F = 0, then F is said to be incompressible.

  11. VECTOR FORMS OFGREEN’S THEOREM Theorem: Let C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C. Suppose P and Q have continuous partial derivatives on an open region that contains D. Consider the vector field F = P i + Qj. Then and

  12. EXAMPLE Use the vector forms of Green’s Theorem to calculate where C is the boundary of the unit square with vertices (0, 0), (1, 0), (1, 1), and (0, 1) where F = 3yi + 5xj.

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