Let k be a constant, F = F( x , y , z ) , G = G( x , y , z ) , and ϕ = ϕ ( x , y , z ) . Prove the following identities, assuming that all derivatives involved exist and are continuous. curl ( ∇ ϕ ) = 0
Let k be a constant, F = F( x , y , z ) , G = G( x , y , z ) , and ϕ = ϕ ( x , y , z ) . Prove the following identities, assuming that all derivatives involved exist and are continuous. curl ( ∇ ϕ ) = 0
Let k be a constant,
F
=
F(
x
,
y
,
z
)
,
G
=
G(
x
,
y
,
z
)
,
and
ϕ
=
ϕ
(
x
,
y
,
z
)
.
Prove the following identities, assuming that all derivatives involved exist and are continuous.
Let r(t) be a vector-valued function such that the magnitude of r(t) does not change over time. Use derivatives to show that the derivative r'(t) is perpendicular to the function r(t) for all times t.
OW + OW = 0 for w = f(x, y), x = u – v, and y = v –- u and demonstrate the
ди ду
6. Show that
result for w = (x – y) sin (y – x).
Let h(x, y) be a differentiable function and let (xo, yo) be a point in the domain of h. Also, let u be a
unit vector. Then
D- h(xo, yo) = -Du h(xo, yo).
Hint: Your job is to show that two directional derivatives are related. First notice the two negative signs.
One of them is the negative of a VECTOR. Start by computing the directional derivative on the LHS.
Remember that for the dot product, a (-b) = (-a) b=-(ab).
Thomas' Calculus: Early Transcendentals (14th Edition)
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY