1. Vector Fields

2. Time Derivative  Derivatives of vectors are by component.  Derivatives of vector products use the chain rule. Scalar multiplicationScalar multiplication Inner productInner product Vector productVector product

3. Normal Vectors Example  Show that the velocity of a particle at constant speed is normal to acceleration.  Use the inner product. Defines magnitude Commutes  Since v i a i = 0, vector v is normal to a.

4. Space Derivative  Spatial derivatives depend on the coordinates.  The partial derivatives point along coordinate lines. Not the same as the coordinates.Not the same as the coordinates.  The del operator is not a vector but acts like one. Gradient changes scalar to vectorGradient changes scalar to vector y = const. x = const. y x

5. Vector Field  A vector field is a vector that depends on position.  The differential operator is a vector field. Acts on a scalar fieldActs on a scalar field Measures changeMeasures change From Wolfram’s Mathworld

6. Divergence  The inner product of the del operator with a vector is the divergence. Scalar resultScalar result  The divergence of a gradient is the Laplacian. Del squared operatorDel squared operator  Divergence is related to the flow from a volume.

7. Curl  The vector product of the del operator with a vector is the curl. Vector result  The divergence of a curl is zero.  Curl is related to the inner product with the tangent vector t. but next