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
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 36E
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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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4fc8276-ad27-42b6-9c31-60320bfac928%2Fc51b03ff-a798-416f-91f0-20a6af5e8e3a%2F0iz6q47_processed.jpeg&w=3840&q=75)
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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