Let f be a differentiable function of one variable, and let w = f ρ , where ρ = x 2 + y 2 + z 2 1 / 2 . show that ∂ w ∂ x 2 + ∂ w ∂ y 2 + ∂ w ∂ z 2 = d w d ρ 2
Let f be a differentiable function of one variable, and let w = f ρ , where ρ = x 2 + y 2 + z 2 1 / 2 . show that ∂ w ∂ x 2 + ∂ w ∂ y 2 + ∂ w ∂ z 2 = d w d ρ 2
Let f be a differentiable function of one variable, and let
w
=
f
ρ
,
where
ρ
=
x
2
+
y
2
+
z
2
1
/
2
.
show that
∂
w
∂
x
2
+
∂
w
∂
y
2
+
∂
w
∂
z
2
=
d
w
d
ρ
2
F being a function of two variables;
F( x= az, y=bz)
z = z(x, y) defined by the relation
a∂z/∂x + b∂z/∂y = 1
Show that it satisfies the equation.
Let f(x, y) = Vx?y.
a. Sketch/describe the domain of g
Domain = { (x, y) ]y 2 0 }
b. Sketch a contour map (level curves on one set of axes) for z = 1,2,3.
https://www.desmos.com/calculator/lqsz6yuywz
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).
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