For the following exercises, without using Stokes’ theorem, calculate directly both the flux of curl F ⋅ N over the given surface and the circulation integral around its boundary, assuming all boundaries are oriented clockwise as viewed from above. 326. F ( x , y , z ) = y 2 i + z 2 j + x 2 k ; S is the first-octant portion of plane x + y + z = 1 .
For the following exercises, without using Stokes’ theorem, calculate directly both the flux of curl F ⋅ N over the given surface and the circulation integral around its boundary, assuming all boundaries are oriented clockwise as viewed from above. 326. F ( x , y , z ) = y 2 i + z 2 j + x 2 k ; S is the first-octant portion of plane x + y + z = 1 .
For the following exercises, without using Stokes’ theorem, calculate directly both the flux of curl
F
⋅
N
over the given surface and the circulation integral around its boundary, assuming all boundaries are oriented clockwise as viewed from above.
326.
F
(
x
,
y
,
z
)
=
y
2
i
+
z
2
j
+
x
2
k
;
S
is the first-octant portion of plane
x
+
y
+
z
=
1
.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
(b) Let I[y] be a functional of y(x) defined by
[[y] = √(x²y' + 2xyy' + 2xy + y²) dr,
subject to boundary conditions
y(0) = 0,
y(1) = 1.
State the Euler-Lagrange equation for finding extreme values of I [y] for this prob-
lem. Explain why the function y(x) = x is an extremal, and for this function,
show that I = 2. Without doing further calculations, give the values of I for the
functions y(x) = x² and y(x) = x³.
Please use mathematical induction to prove this
In simplest terms, Sketch the graph of the parabola. Then, determine its equation.
opens downward, vertex is (- 4, 7), passes through point (0, - 39)
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