Proof Let r( t ) and u( t ) be vector -valued functions whose limits exist as t → c . Prove that lim x → c [ r ( t ) ⋅ u ( t ) ] = lim x → c r ( t ) ⋅ lim x → c u ( t ) .
Solution Summary: The author explains the limit of r(t), and the formula used: the dot product of two vectors.
Proof Let r(t) and u(t) be vector-valued functions whose limits exist as
t
→
c
. Prove that
lim
x
→
c
[
r
(
t
)
⋅
u
(
t
)
]
=
lim
x
→
c
r
(
t
)
⋅
lim
x
→
c
u
(
t
)
.
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.
Identities Prove the following identities. Assume φ is a differentiablescalar-valued function and F and G are differentiable vectorfields, all defined on a region of ℝ3.
∇ x (∇ x F) = ∇(∇ ⋅ F) - (∇ ⋅ ∇)F
Identities Prove the following identities. Assume φ is a differentiablescalar-valued function and F and G are differentiable vectorfields, all defined on a region of ℝ3.
∇ ⋅ (F x G ) = G ⋅ (∇ x F) - F ⋅ (∇ x G)
Identities Prove the following identities. Assume φ is a differentiablescalar-valued function and F and G are differentiable vectorfields, all defined on a region of ℝ3.
∇ (F ⋅ G ) = (G ⋅ ∇) F + (F ⋅ ∇)G + G x (∇ x F) + F x (∇ x G)
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