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Calculus Volume 3

Key Terms

double integral
of the function $f(x,y)f(x,y)$ over the region $RR$ in the $xyxy$-plane is defined as the limit of a double Riemann sum, $∬Rf(x,y)dA=limm,n→∞∑i=1m∑j=1nf(xij*,yij*)ΔA.∬Rf(x,y)dA=limm,n→∞∑i=1m∑j=1nf(xij*,yij*)ΔA.$
double Riemann sum
of the function $f(x,y)f(x,y)$ over a rectangular region $RR$ is $∑i=1m∑j=1nf(xij*,yij*)ΔA∑i=1m∑j=1nf(xij*,yij*)ΔA$ where $RR$ is divided into smaller subrectangles $RijRij$ and $(xij*,yij*)(xij*,yij*)$ is an arbitrary point in $RijRij$
Fubini’s theorem
if $f(x,y)f(x,y)$ is a function of two variables that is continuous over a rectangular region $R={(x,y)∈ℝ2|a≤x≤b,c≤y≤d},R={(x,y)∈ℝ2|a≤x≤b,c≤y≤d},$ then the double integral of $ff$ over the region equals an iterated integral, $∬Rf(x,y)dydx=∫ab∫cdf(x,y)dxdy=∫cd∫abf(x,y)dxdy∬Rf(x,y)dydx=∫ab∫cdf(x,y)dxdy=∫cd∫abf(x,y)dxdy$
improper double integral
a double integral over an unbounded region or of an unbounded function
iterated integral
for a function $f(x,y)f(x,y)$ over the region $RR$ is
1. $∫ab∫cdf(x,y)dxdy=∫ab[∫cdf(x,y)dy]dx,∫ab∫cdf(x,y)dxdy=∫ab[∫cdf(x,y)dy]dx,$
2. $∫cd∫baf(x,y)dxdy=∫cd[∫abf(x,y)dx]dy,∫cd∫baf(x,y)dxdy=∫cd[∫abf(x,y)dx]dy,$
where $a,b,c,a,b,c,$ and $dd$ are any real numbers and $R=[a,b]×[c,d]R=[a,b]×[c,d]$
Jacobian
the Jacobian $J(u,v)J(u,v)$ in two variables is a $2×22×2$ determinant:
$J(u,v)=|∂x∂u∂y∂u∂x∂v∂y∂v|;J(u,v)=|∂x∂u∂y∂u∂x∂v∂y∂v|;$

the Jacobian $J(u,v,w)J(u,v,w)$ in three variables is a $3×33×3$ determinant:
$J(u,v,w)=|∂x∂u∂y∂u∂z∂u∂x∂v∂y∂v∂z∂v∂x∂w∂y∂w∂z∂w|J(u,v,w)=|∂x∂u∂y∂u∂z∂u∂x∂v∂y∂v∂z∂v∂x∂w∂y∂w∂z∂w|$
one-to-one transformation
a transformation $T:G→RT:G→R$ defined as $T(u,v)=(x,y)T(u,v)=(x,y)$ is said to be one-to-one if no two points map to the same image point
planar transformation
a function $TT$ that transforms a region $GG$ in one plane into a region $RR$ in another plane by a change of variables
polar rectangle
the region enclosed between the circles $r=ar=a$ and $r=br=b$ and the angles $θ=αθ=α$ and $θ=β;θ=β;$ it is described as $R={(r,θ)|a≤r≤b,α≤θ≤β}R={(r,θ)|a≤r≤b,α≤θ≤β}$
radius of gyration
the distance from an object’s center of mass to its axis of rotation
transformation
a function that transforms a region $GG$ in one plane into a region $RR$ in another plane by a change of variables
triple integral
the triple integral of a continuous function $f(x,y,z)f(x,y,z)$ over a rectangular solid box $BB$ is the limit of a Riemann sum for a function of three variables, if this limit exists
triple integral in cylindrical coordinates
the limit of a triple Riemann sum, provided the following limit exists:
$liml,m,n→∞∑i=1l∑j=1m∑k=1nf(rijk*,θijk*,zijk*)rijk*ΔrΔθΔzliml,m,n→∞∑i=1l∑j=1m∑k=1nf(rijk*,θijk*,zijk*)rijk*ΔrΔθΔz$
triple integral in spherical coordinates
the limit of a triple Riemann sum, provided the following limit exists:
$liml,m,n→∞∑i=1l∑j=1m∑k=1nf(ρijk*,θijk*,φijk*)(ρijk*)2sinφΔρΔθΔφliml,m,n→∞∑i=1l∑j=1m∑k=1nf(ρijk*,θijk*,φijk*)(ρijk*)2sinφΔρΔθΔφ$
Type I
a region $DD$ in the $xyxy$-plane is Type I if it lies between two vertical lines and the graphs of two continuous functions $g1(x)g1(x)$ and $g2(x)g2(x)$
Type II
a region $DD$ in the $xyxy$-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions $h1(y)andh2(y)h1(y)andh2(y)$
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