Another derivative combination Let F = ( f, g, h ) and let u be a differentiable scalar-valued function. a. Take the dot product of F and the del operator; then apply the result to u to show that ( F ⋅ ∇ ) u = ( f ∂ ∂ x + g ∂ ∂ y + h ∂ ∂ z ) u = f ∂ u ∂ x + g ∂ u ∂ y + h ∂ u ∂ z . b. Evaluate ( F · ▿) ( xy 2 z 3 ) at (1, 1 , 1), where F = (1 , 1, 1).
Another derivative combination Let F = ( f, g, h ) and let u be a differentiable scalar-valued function. a. Take the dot product of F and the del operator; then apply the result to u to show that ( F ⋅ ∇ ) u = ( f ∂ ∂ x + g ∂ ∂ y + h ∂ ∂ z ) u = f ∂ u ∂ x + g ∂ u ∂ y + h ∂ u ∂ z . b. Evaluate ( F · ▿) ( xy 2 z 3 ) at (1, 1 , 1), where F = (1 , 1, 1).
Another derivative combination Let F = (f, g, h) and let u be a differentiable scalar-valued function.
a. Take the dot product of F and the del operator; then apply the result to u to show that
(
F
⋅
∇
)
u
=
(
f
∂
∂
x
+
g
∂
∂
y
+
h
∂
∂
z
)
u
=
f
∂
u
∂
x
+
g
∂
u
∂
y
+
h
∂
u
∂
z
.
b. Evaluate (F·▿)(xy2z3) at (1, 1, 1), where F = (1, 1, 1).
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Describe what it means for a vector-valued function r(t) to be continuous at a point.
Express
∂w∂u
and
∂w∂v
as functions of u and v.
Another derivative combination Let F = (f. g, h) and let u be
a differentiable scalar-valued function.
a. Take the dot product of F and the del operator; then apply the
result to u to show that
(F•V )u = (3
a
+ h
az
(F-V)u
+ g
+ g
du
+ h
b. Evaluate (F - V)(ry²z³) at (1, 1, 1), where F = (1, 1, 1).
Chapter 17 Solutions
Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
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Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY