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POST SECONDARY
CALCULUS AND ANALYSIS
MULTIVARIABLE CALCULUS
Secondary
Post Secondary
Algebra
Statistics and Probability
Calculus and Analysis
Integral Calculus
Multivariable calculus
Multivariable functions
Green's, Stokes', and the divergence theorem
Differential Calculus
Analysis
Differential equations
Advanced Math
Differential equations
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Multivariable calculus Answers
Multivariable functions
asked 2021-03-18
Write formulas for the indicated partial derivatives for the multivariable function.
\(\displaystyle{f{{\left({x},{y}\right)}}}={7}{x}^{{2}}+{9}{x}{y}+{4}{y}^{{3}}\)
a)
\(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}\)
b)(delf)/(dely)ZSK
c)
\(\displaystyle\frac{{\partial{f}}}{{\partial{x}}}{\mid}_{{{y}={9}}}\)
Green's, Stokes', and the divergence theorem
asked 2021-03-12
Use Green's Theorem to evaluate the line integral
\(\displaystyle\int_{{C}}{\left({y}+{e}^{{x}}\right)}{\left.{d}{x}\right.}+{\left({6}{x}+{\cos{{y}}}\right)}{\left.{d}{y}\right.}\)
where C is triangle with vertices (0,0),(0,2)and(2,2) oriented counterclockwise.
a)6
b)10
c)14
d)4
e)8
f)12
Green's, Stokes', and the divergence theorem
asked 2021-03-11
Use the divergence theorem to evaluate
\(\displaystyle\int\int_{{S}}{F}\cdot{N}{d}{S}\)
, where
\(\displaystyle{F}{\left({x},{y},{z}\right)}={y}^{{2}}{z}{i}+{y}^{{3}}{j}+{x}{z}{k}\)
and S is the boundary of the cube defined by
\(\displaystyle-{5}\le{x},={5},-{5}\le{y}\le{5},{\quad\text{and}\quad}{0}\le{z}\le{10}\)
.
Green's, Stokes', and the divergence theorem
asked 2021-03-09
Use Stokes' Theorem to evaluate
\(\displaystyle\int\int_{{S}}{C}{U}{R}{L}{f}\cdot{d}{S}\)
.
\(\displaystyle{F}{\left({x},{y},{z}\right)}={x}^{{2}}{y}^{{3}}{z}{i}+{\sin{{\left({x}{y}{z}\right)}}}{j}+{x}{y}{z}{k}\)
,
S is the part of the cone
\(\displaystyle{y}^{{2}}={x}^{{2}}+{z}^{{2}}\)
that lies between the planes y = 0 and y = 2, oriented in the direction of the positive y-axis.
Green's, Stokes', and the divergence theorem
asked 2021-03-08
z = x Let be the curve of intersection of the cylinder
\(\displaystyle{x}^{{2}}+{y}^{{2}}={1}\)
and the plane , oriented positively when viewed from above . Let S be the inside of this curve , oriented with upward -pointing normal . Use Stokes ' Theorem to evaluate
\(\displaystyle\int{S}{c}{u}{r}{l}{F}\cdot{d}{S}{\quad\text{if}\quad}{F}={y}{i}+{z}{j}+{2}{x}{k}\)
.
Green's, Stokes', and the divergence theorem
asked 2021-03-05
Apply Green’s theorem to find the outward flux for the field
\(\displaystyle{F}{\left({x},{y}\right)}={{\tan}^{{−{1}}}{\left(\frac{{y}}{{x}}\right)}}{i}+{\ln{{\left({x}^{{2}}+{y}^{{2}}\right)}}}{j}\)
Green's, Stokes', and the divergence theorem
asked 2021-03-04
Use Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity grad g · n = Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)
\(\displaystyle\oint_{{c}}{F}\cdot{n}{d}{s}=\int\int_{{D}}\div{F}{\left({x},{y}\right)}{d}{A}\)
Multivariable functions
asked 2021-03-02
COnsider the multivariable function
\(\displaystyle{g{{\left({x},{y}\right)}}}={x}^{{2}}-{3}{y}^{{4}}{x}^{{2}}+{\sin{{\left({x}{y}\right)}}}\)
. Find the following partial derivatives:
\(\displaystyle{g}_{{x}}.{g}_{{y}},{g}_{{{x}{y}}},{g{{\left(\times\right)}}},{g{{\left({y}{y}\right)}}}\)
.
Multivariable functions
asked 2021-03-01
Write first and second partial derivatives
\(\displaystyle{g{{\left({r},{t}\right)}}}={t}{\ln{{r}}}+{11}{r}{t}^{{7}}-{5}{\left({8}^{{r}}\right)}-{t}{r}\)
a)
\(\displaystyle{g}_{{r}}\)
b)
\(\displaystyle{g}_{{{r}{r}}}\)
c)
\(\displaystyle{g}_{{{r}{t}}}\)
d)
\(\displaystyle{g}_{{t}}\)
e)
\(\displaystyle{g}_{{{\mathtt}}}\)
Green's, Stokes', and the divergence theorem
asked 2021-03-01
Use Green's Theorem to evaluate
\(\displaystyle\int_{{C}}\vec{{{F}}}\cdot{d}\vec{{{r}}}\)
where
\(\displaystyle\vec{{{F}}}{\left({x},{y}\right)}={x}{y}^{{2}}{i}+{\left({1}-{x}{y}^{{3}}\right)}{j}\)
and C is the parallelogram with vertices (-1,2), (-1,-1),(1,1)and(1,4).
The orientation of C is counterclockwise.
Multivariable functions
asked 2021-02-27
Write formulas for the indicated partial derivatives for the multivariable function.
\(\displaystyle{g{{\left({k},{m}\right)}}}={k}^{{3}}{m}^{{6}}−{8}{k}{m}\)
a)
\(\displaystyle{g}_{{k}}\)
b)
\(\displaystyle{g}_{{m}}\)
c)
\(\displaystyle{g}_{{m}}{\mid}_{{{k}={2}}}\)
Green's, Stokes', and the divergence theorem
asked 2021-02-25
Use Green's Theorem to evaluate
\(\displaystyle\int_{{C}}{\left({e}^{{x}}+{y}^{{2}}\right)}{\left.{d}{x}\right.}+{\left({e}^{{y}}+{x}^{{2}}\right)}{\left.{d}{y}\right.}\)
where C is the boundary of the region(traversed counterclockwise) in the first quadrant bounded by
\(\displaystyle{y}={x}^{{2}}{\quad\text{and}\quad}{y}={4}\)
.
Green's, Stokes', and the divergence theorem
asked 2021-02-25
Let
\(\displaystyle{F}{\left({x},{y}\right)}={\left\langle{4}{\cos{{\left({y}\right)}}},{2}{\sin{{\left({y}\right)}}}\right\rangle}\)
. Compute the flux
\(\displaystyle\oint{F}\cdot{n}{d}{s}\)
of F across the boundary of the rectangle
\(\displaystyle{0}\le{x}\le{5},{0}\le{y}\le\frac{\pi}{{2}}\)
using the vector form of Green's Theorem.
\(\displaystyle\oint{F}\cdot{n}{d}{s}=\)
?
Green's, Stokes', and the divergence theorem
asked 2021-02-25
Use Stokes' Theorem to evaluate
\(\displaystyle\int_{{C}}{F}\cdot{d}{r}\)
where C is oriented counterclockwise as viewed from above.
\(\displaystyle{F}{\left({x},{y},{z}\right)}={\left({x}+{y}^{{2}}\right)}{i}+{\left({y}+{z}^{{2}}\right)}{j}+{\left({z}+{x}^{{2}}\right)}{k}\)
,
C is the triangle with vertices (3,0,0),(0,3,0), and (0,0,3).
Green's, Stokes', and the divergence theorem
asked 2021-02-25
Evaluate
\(\displaystyle\int_{{C}}{x}^{{2}}{y}^{{2}}{\left.{d}{x}\right.}+{4}{x}{y}^{{3}}{\left.{d}{y}\right.}\)
where C is the triangle with vertices(0,0),(1,3), and (0,3).
(a)Use the Green's Theorem.
(b)Do not use the Green's Theorem.
Multivariable functions
asked 2021-02-22
Write formulas for the indicated partial derivatives for the multivariable function.
\(\displaystyle{g{{\left({x},{y},{z}\right)}}}={3.1}{x}^{{2}}{y}{z}^{{2}}+{2.7}{x}^{{y}}+{z}\)
a)
\(\displaystyle{g}_{{x}}\)
b)
\(\displaystyle{g}_{{y}}\)
c)
\(\displaystyle{g}_{{z}}\)
Multivariable functions
asked 2021-02-22
Multivariable optimization. Find the demisions of the rectangular box with largest volume if the total surface are is given as 64
\(\displaystyle{c}{m}^{{2}}\)
Green's, Stokes', and the divergence theorem
asked 2021-02-21
Use Green's Theorem to evaluate
\(\displaystyle\oint_{{C}}{\left({x}^{{2}}+{y}\right)}{\left.{d}{x}\right.}-{\left({3}{x}+{y}^{{3}}\right)}{\left.{d}{y}\right.}\)
Where c is the ellipse
\(\displaystyle{x}^{{2}}+{4}{y}^{{2}}={4}\)
Green's, Stokes', and the divergence theorem
asked 2021-02-21
If E(t,x,y,z) and B(t,x,y,z)represent the electric and magnetic fields at point (x,y,z) at time t, a basic principle of electromagnetic theory says that
\(\displaystyle\nabla\times{E}=\frac{{-\partial{B}}}{{\partial{t}}}\)
. In this expression
\(\displaystyle\nabla\times{E}\)
is computed with t held fixed and
\(\displaystyle\frac{{\partial{B}}}{{\partial{t}}}\)
is calculated with (x,y,z) fixed.
Use Stokes' Theorem to derive Faraday's law,
\(\displaystyle\oint_{{C}}{E}\cdot{d}{r}=-\frac{\partial}{{\partial{t}}}\int\int_{{S}}{B}\cdot{n}{d}\sigma\)
,
Green's, Stokes', and the divergence theorem
asked 2021-02-21
Let
\(\displaystyle{F}={\left[{x}^{{2}},{0},{z}^{{2}}\right]}\)
, and S the surface of the box
\(\displaystyle{\left|{{x}}\right|}\le{1},{\left|{{y}}\right|}\le{3},{0}\le{z}\le{2}\)
.
Evaluate the surface integral
\(\displaystyle\int\int_{{S}}{F}\cdot{n}{d}{A}\)
by the divergence theorem.
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