MATH-M 312 Lecture Notes - Lecture 13: Differentiable Function, Scalar Multiplication, Product Rule
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M312 section 4. 4 notes- gradients, divergence, and curl. 3-28-16: formal properties, formal properties, physical interpretation. Gradient- takes scalar fields to vector fields. Divergence- takes vector fields to scalar fields. =(cid:4672)(cid:3051),(cid:3052),(cid:3053)(cid:4673)=((cid:3051),(cid:3052),(cid:3053)) is vector of differential operators, not real numbers. (cid:1858)=(cid:4672)(cid:3051),(cid:3052),(cid:3053)(cid:4673)(cid:1858)- treat like scalar multiplication to get (cid:1858)=(cid:4672)(cid:3051),(cid:3052),(cid:3053)(cid:4673) All 3 differe(cid:374)tial operators (cid:272)a(cid:374) (cid:271)e defi(cid:374)ed i(cid:374) ter(cid:373)s of (cid:862)del(cid:863) (cid:4666) (cid:4667) Gradient field- any vector field that is the gradient of some scalar field; (cid:2163) is gradient field if (cid:2163)= (cid:2162) Gradient vector points in direction of maximum increase of (cid:1858) at. | (cid:1858)(cid:4666)(cid:4667)|= rate of i(cid:374)(cid:272)rease/(cid:862)slope(cid:863) of graph of (cid:1858) at in direction of maximum increase: ex. (cid:1858)(cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667)=(cid:1876)(cid:2870)+(cid:1877)(cid:1878)+(cid:1876)(cid:1877)(cid:1878) Since (cid:2162) is vector field, we say (cid:2162)=(cid:4666)(cid:1832)(cid:2869),(cid:1832)(cid:2870),(cid:1832)(cid:2871)(cid:4667) where (cid:1832)(cid:2869),(cid:1832)(cid:2870),(cid:1832)(cid:2871) are scalar fields. Divergence field- any scalar field which is the divergence of some vector field; (cid:2163) is divergence field if (cid:2163)= (cid:2162) =||(cid:3051) (cid:3052) (cid:3053) (cid:1832)(cid:2869) (cid:1832)(cid:2870) (cid:1832)(cid:2871)|| (cid:3053) (cid:3052) (cid:3051) =(cid:882) by equality of mixed partials for differentiable functions.
