Consider the surface integral ∬ σ f x , y , z d S . (a) If σ is a parametric surface whose vector equation is r = x u , v i + y u , v j + z u , v k to evaluate the integral replace d S by ______ . (b) If σ is the graph of a function z = g x , y with continuous first partial derivatives, to evaluate the integral replace d S by ______ .
Consider the surface integral ∬ σ f x , y , z d S . (a) If σ is a parametric surface whose vector equation is r = x u , v i + y u , v j + z u , v k to evaluate the integral replace d S by ______ . (b) If σ is the graph of a function z = g x , y with continuous first partial derivatives, to evaluate the integral replace d S by ______ .
Consider the surface integral
∬
σ
f
x
,
y
,
z
d
S
.
(a) If
σ
is a parametric surface whose vector equation is
r
=
x
u
,
v
i
+
y
u
,
v
j
+
z
u
,
v
k
to evaluate the integral replace
d
S
by
______
.
(b) If
σ
is the graph of a function
z
=
g
x
,
y
with continuous first partial derivatives, to evaluate the integral replace
d
S
by
______
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
The vector v = <a, 1, -1>, is tangent to the surface x2 + 2y3 - 3z2 = 3 at the point (2, 1, 1).
Find a.
4. Consider the vector function r(z, y) (r, y, r2 +2y").
(a) Re-write this vector function as surface function in the form f(1,y).
(b) Describe and draw the shape of the surface function using contour lines and algebraic analysis
as needed. Explain the contour shapes in all three orthogonal directions and explain and label
all intercepts as needed.
(c) Consider the contour of the surface function on the plane z=
for this contour in vector form.
0. Write the general equation
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: Mathematics for Calculus - 6th Edition
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