Determine whether the statement is true or false. Explain your answer. Suppose that z = f x , y has continuous first partial derivatives in the interior of a region R in the x y -plane , and set q = 1 , 0 , ∂ z / ∂ x and r = 0 , 1 , ∂ z / ∂ y . Then the surface area of the surface z = f x , y over R is ∬ R q × r d A
Determine whether the statement is true or false. Explain your answer. Suppose that z = f x , y has continuous first partial derivatives in the interior of a region R in the x y -plane , and set q = 1 , 0 , ∂ z / ∂ x and r = 0 , 1 , ∂ z / ∂ y . Then the surface area of the surface z = f x , y over R is ∬ R q × r d A
Determine whether the statement is true or false. Explain your answer.
Suppose that
z
=
f
x
,
y
has continuous first partial derivatives in the interior of a region R in the
x
y
-plane
,
and set
q
=
1
,
0
,
∂
z
/
∂
x
and
r
=
0
,
1
,
∂
z
/
∂
y
.
Then the surface area of the surface
z
=
f
x
,
y
over R is
Find the area of the surface defined by x + y + z = 1, x2 + 2y2 ≤ 1.
H. Use the gradient to find the equation of the tangent plane to each of the surfaces at the given point.
a) x² + 3x²y-z = 0 at (1,1,4) (Answ: 9x+3y-z = 8)
b) z = f(x, y, z) = r²y³z at (2,1,3) (Answ: 4x - 3y -z = 2)
I. In electrostatics the force (F) of attraction between two particles of opposite charge is given by
(Coulomb's law) where k is a constant and r = (x, y, z). Show that F is the gradient
T
(Hint: ||||||(x, y, z)||). Important problem!
F(r) = k₁
of P(7)
||7-1³
-k
||1|
=
az.
Suppose F = (2xz + 3y²) a, + (4yz²) a;.
(a) Calculate S[F·dS, where S is the shaded surface in Figure 1.
(c) Based on your results for parts (a) and (b), what named theorem do you think is
being satisfied here, if any?
(b) Calculate SF· dl, where C is the A → B → C → D → A closed path in Figure 1.
az
C
C (0,1,1)
D
(0,0,0)
(A
ay
В
ax
Figure 1: Figure for Problem 1.
Calculus, Single Variable: Early Transcendentals (3rd Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY