Maximum surface integral Let S be the paraboloid z = a (1 – x 2 – y 2 ) , for z ≥ 0, where a > 0 is a real number. Let F = ( x – y , y + z, z – x ) . For what value(s) of a (if any) does ∬ S ( ∇ × F ) ⋅ n d S have its maximum value?
Maximum surface integral Let S be the paraboloid z = a (1 – x 2 – y 2 ) , for z ≥ 0, where a > 0 is a real number. Let F = ( x – y , y + z, z – x ) . For what value(s) of a (if any) does ∬ S ( ∇ × F ) ⋅ n d S have its maximum value?
Maximum surface integral Let S be the paraboloid z = a(1 – x2– y2), for z ≥ 0, where a > 0 is a real number. Let F = (x – y, y + z, z – x). For what value(s) of a (if any) does
∬
S
(
∇
×
F
)
⋅
n
d
S
have its maximum value?
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.
1. Consider the function F(x, y, z) = (√/1 – x² − y², ln(e² — z²)).
This function is a mapping from R" to Rm. Determine the values of m and n.
(b) Is this function scalar-valued or vector-valued? Briefly explain.
(c) Determine the domain and range of F and sketch the corresponding regions.
(d) Is it possible to visualize this function as a graph? If so, sketch the graph of F.
Assume that u and u are continuously differentiable functions. Using Green's theorem,
prove that
Uz
JE v|dA= [udv,
Uy
D
where D is some domain enclosed by a simple closed curve C with positive orientation.
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