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Theorem 14. Properties of the Gradient Let f and g both be real-valued functions of two or three variables that are differentiable on a common open set O. 1. Sum Rule 2. Constant Multiple Rule 3. Product Rule v(f+g) = Vf+Vg (cf)=cf for all ceR v(fg) =fvg+gVf

Theorem 14. Properties of the Gradient Let f and g both be real-valued functions of two or three variables that are differentiable on a common open set O. 1. Sum Rule 2. Constant Multiple Rule 3. Product Rule v(f+g) = Vf+Vg (cf)=cf for all ceR v(fg) =fvg+gVf

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove parts 1, 2, and 3 of Theorem 14.

Theorem 14. Properties of the Gradient Let f and g both be real-valued functions of two or three variables that are differentiable on a common open set O. 1. Sum Rule 2. Constant Multiple Rule 3. Product Rule v(f+g) = Vf+Vg (cf)=cf for all ceR v(fg) =fvg+gVf 4. Chain Rule If h is a differentiable real-valued function of one variable defined on a domain D, and if the range of f satisfies R, D = Dw, then v(hof) = (h' of)vf
Transcribed Image Text:Theorem 14. Properties of the Gradient Let f and g both be real-valued functions of two or three variables that are differentiable on a common open set O. 1. Sum Rule 2. Constant Multiple Rule 3. Product Rule v(f+g) = Vf+Vg (cf)=cf for all ceR v(fg) =fvg+gVf 4. Chain Rule If h is a differentiable real-valued function of one variable defined on a domain D, and if the range of f satisfies R, D = Dw, then v(hof) = (h' of)vf
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