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[SADT9] The Laplacian of a function f of n variables x 1 ,x 2 ,*** x n denoted nabla^ 2 f is defined by
nabla^ 2 f(x 1 ,x 2 ,***,x n ):= partial^ 2 f partial x 1 ^ 2 + partial^ 2 f partial x 2 ^ 2 +***+ partial^ 2 f partial x n ^ 2
Now assume that f depends only on r where r=(x 1 ^ 2 +x 2 ^ 2 +***+x n ^ 2 )^ 1 2 i.e. f(x 1 ,x 2 ,***,x n )=g(r) for some function g. Show that, for x 1 ,x 2 ,***,x n ne0 ,
nabla^ 2 f= n-1 r g^ prime (r)+g^ prime prime (r)
[SADT3] For scalar functions u and v, show that
B=( nabla u)*( nabla v)
is solenoidal and that
A= 1 2 (u nabla v-v nabla u)
is a vector potential for B, i.e. B= nabla* A
With respect to the bar of chocolate, where is their center of mass?
The initial x-coordinates of James and Ramon are -10.0 m and +10.0 m respectively,
so the x-coordinate of the center of mass is:
Formula:
Solution:
Final Answer
The Laplacian of a function f
of n
variables x
1
,x
2
,⋯x
n
, denoted ∇
2
f
is defined by
∇
2
f(x
1
,x
2
,⋯,x
n
):=∂
2
f
∂x
2
1
+∂
2
f
∂x
2
2
+⋯+∂
2
f
∂x
2
n
Now assume that f
depends only on r
where r=(x
2
1
+x
2
2
+⋯+x
2
n
)
1
2
, i.e. f(x
1
,x
2
,⋯,x
n
)=g(r)
, for some function g
. Show that, for x
1
,x
2
,⋯,x
n
≠0
,
∇
2
f=n−1
r
g
′
(r)+g
′′
(r)
If A
and B
are vector fields, prove the following:
∇(A⋅B)=(B⋅∇)A+(A⋅∇)B+B×(∇×A)+A×(∇×B).
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Find the solution of the recurrence relation an = 4an−1 − 3an−2 + 2^n + n + 3 with
a^0 = 1 and a^1 = 4.
- D2((x+1)2/x-1) ans, 8/(x-1)3
- D(ln(x2-2x+1)) ans, 2/x-1
- D(x2+2x-3) ans, 2x+2
- D2(xe3x-e4x) ans, 9xe3x+633x-16e4x
Solve 7x+9y=3
5x+7y=1 using matrix method
Jason estimates that his car loses 12% of its value every year. The initial value is $12,000. Which best describes the graph of the function that represents the value of the car after x years?
f(x) = 12,000(0.12)x, with a horizontal asymptote of y = 0
f(x) = 12,000(1.12)x, with a vertical asymptote of x = 0
f(x) = 12,000(0.88)x, with a horizontal asymptote of y = 0
f(x) = (12,000 0.88)x, with a vertical asymptote of x = 0
