(a) Consider the transformation x = r cos θ , y = r sin θ , z = z from cylindrical to rectangular coordinates where r ≥ 0. Show that ∂ x , y , z ∂ r , θ , z = r (b) Consider the transformation x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ from spherical to rectangular coordinates, where 0 ≤ ϕ ≤ π . Show that ∂ x , y , z ∂ ρ , ϕ , θ = ρ 2 sin ϕ
(a) Consider the transformation x = r cos θ , y = r sin θ , z = z from cylindrical to rectangular coordinates where r ≥ 0. Show that ∂ x , y , z ∂ r , θ , z = r (b) Consider the transformation x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ from spherical to rectangular coordinates, where 0 ≤ ϕ ≤ π . Show that ∂ x , y , z ∂ ρ , ϕ , θ = ρ 2 sin ϕ
(a) Consider the transformation
x
=
r
cos
θ
,
y
=
r
sin
θ
,
z
=
z
from cylindrical to rectangular coordinates where
r
≥
0.
Show that
∂
x
,
y
,
z
∂
r
,
θ
,
z
=
r
(b) Consider the transformation
x
=
ρ
sin
ϕ
cos
θ
,
y
=
ρ
sin
ϕ
sin
θ
,
z
=
ρ
cos
ϕ
from spherical to rectangular coordinates, where
0
≤
ϕ
≤
π
.
Show that
∂
x
,
y
,
z
∂
ρ
,
ϕ
,
θ
=
ρ
2
sin
ϕ
Find both parametric and rectangular representations for the plane tangent to r(u,v)=u2i+ucos(v)j+usin(v)kr(u,v)=u2i+ucos(v)j+usin(v)k at the point P(4,−2,0)P(4,−2,0).One possible parametric representation has the form⟨4−4u⟨4−4u , , 4v⟩4v⟩(Note that parametric representations are not unique. If your first and third components look different than the ones presented here, you will need to adjust your parameters so that they do match, and then the other components should match the ones expected here as well.)The equation for this plane in rectangular coordinates has the form x+x+ y+y+ z+z+ =0
please help me
Vector F is mathematically defined as F = M x N, where M = p 2p² cos + 2p2 sind while N
is a vector normal to the surface S. Determine F as well as the area of the plane perpendicular
to F if surface S = 2xy + 3z.
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
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