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Unformatted text preview: Section 17.5 Curl and Divergence New Tools for Line Integrals In the last section, we used the 2-d curl to transform a line integral over a closed curve into a double integral over the region inside the curve. We also used the 2-d curl to determine whether a vector field was conservative. In this section we consider two new functions (one a scalar function and the other a vector function) which we shall be able to use to transform complicated vector integrals into much more straightforward integrals over regions and determine whether a vector field is conservative. 1. Curl We start with a definition. Definition 1.1. Suppose vector F = P vector i + Q vector j + R vector k is a differentiable vector field. Then we define the curl of vector F as the vector function curl( vector F ) = parenleftbigg R y Q z parenrightbigg vector i + parenleftbigg P z R x parenrightbigg vector j + parenleftbigg Q x P y parenrightbigg vector k. The formula for the curl is very long, so it would be useful to have a was to remember it. This can be done through a cross product. Specifically, if we define = x...
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