oldFinalReview - FINAL REVIEW SHEET - MATH 53 ANTON...

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Unformatted text preview: FINAL REVIEW SHEET - MATH 53 ANTON GERASCHENKO This review sheet covers things from after the second midterm. To review the earlier material, look at the Midterm 2 review sheets. Vector Calculus: Derivatives: Youve learned three types of derivatives. The notation is sup- posed to help you remember the formulas. You think of the operator as the vector ( x , y , z ) . (Gradient) You apply gradient to a function to get a vector field: f := (bigg f x , f y , f z )bigg = ( f x , f y , f z ) The direction of the gradient is the direction of maximal increase of f . To get the rate of change of f in the direction of a unit vector u , you take D u f = u f . Perhaps the most useful fact about the gradient is that it is always normal to the level curves or level surfaces of f . (Curl) You apply curl to a vector field to get another vector field: F := vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle i j k /x /y /z P Q R vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = ( R y Q z , P z R x , Q x P y ) where F = ( P, Q, R ) . If you think of the vector field F as the flow of water, then the magnitude of the curl tells you how fast a little paddle wheel would turn if you placed it at that point, and the direction of the curl tells you the axis of rotation of the paddle wheel. To determine the curl of a vector field at a point from a picture, you should draw a little circle around the point of interest and ask, does the vector field push me around this path? If it does, then the curl is non-zero, and you can figure out the direction using the right hand rule. (Divergence) You apply divergence to a vector field to get a function: F := P x + Q y + R z = P x + Q y + R z The divergence measures how much the vector field F is exploding near a point. To determine the divergence of a vector field at a point from a picture, think of the vector field as the flow of water. Put a little blob around the point and ask, does more water leave the blob than comes into it? If more water leaves the blob than comes into it, then the divergence...
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oldFinalReview - FINAL REVIEW SHEET - MATH 53 ANTON...

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