rjacuna

(defn powerset [X]
  (let [n (count X)]
  (->> (range (inc n))
       (map #(clojure.math.combinatorics/combinations (vec X) %))/
       (apply concat)
       (map set)
       (into #{}))))

rjacuna

Surface Area

S is a surface

z=f(x y)
grad(f) exists

Divide D into small rectangles Rij with area dA
Pij(xi,yi,f(xi,yi)) is a point in S. 

rjacuna

14.6 5
Find the directional derivative of f at the given point in the direction indicated by the angle θ.
f(x, y) = y cos(xy),    (0, 1),    θ = π/6

fx=-y^2sin(xy)
fx(0 1)= -1^2sin(0*1) =0

fy= cos(xy)-sin(xy)y^2
fy= 1

rjacuna

14.5 39

The length ℓ, width w, and height h of a box change with time. At a certain instant the dimensions are ℓ = 8 m and 
w = h = 1 m, and ℓ and w are increasing at a rate of 2 m/s while h is decreasing at a rate of 4 m/s. 
At that instant find the rates at which the following quantities are changing.
l = 8 w = 1 h =1
dl = 2 dw = 2 dh = -4

v=lwh
dv = wh dl + lh dw + lw dh

rjacuna

r  <9t t^2 t^3> (9,1,1) => t=1
r' <9 2t 3t^2>
r'(1) <9 2 3>
|r'(1)| sqrt(81+4+9) = sqrt(94)
|r'(1)|^3= 94sqrt(94)
r" <0 2 6t>
r"(1) <0 2 6>

i j k i j
9 2 3 9 2

rjacuna

13.1 19

Find a vector equation and parametric equations for the line segment that joins P to Q.
P(0, −1, 4),    Q(1/2,1/3,1/4)

r(t) = (1-t)r0+tr1

r(t) = (1-t)<0,-1,4>+t<1/2,1/3,1/4>
r(t) = <0,-1,4> -t<0,-1,4>+ t<1/2,1/3,1/4>
r(t) = <0 -0t+1/2t, -1 +t+1/3t, 4-4t+1/4t>

rjacuna

12.4 35
P(-2,1,0) Q(2,2,2) R(1,4,-1) S(3,6,4)

PQ = <2-(-2), 2-1, 2-0> = <4, 1, 2>
PR =<1-(-2), 4-1, -1-0> = <3, 3, -1>
PS =<3-(-2), 6-1, 4-0>  = <5, 5, 4>

Triple Product
     30 -20 12
4 1 2  4 1