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The Euclidean distance between two points x = (x₁, x₂, … , x_n) and x = (y₁, y₂, … , y_n) in Rⁿ is defined by

Px-yP º d(x,y) = [S_{1£ j£ n} (x_j - y_j)²]^1/2.

An open ball or a ball in Rⁿ with center a and radius r is defined by B(a, r) = {x : Px-aP < r}. B[a, r] = {x : Px-aP £ r} is the corresponding closed ball. A ball is also called a sphere and one talks about a closed sphere and an open sphere. In geometry a sphere is used both for the surface S(a, r) = {x : Px-aP = r} of the sphere and the closed sphere B[a, r]. We will use the word sphere or a ball to mean an open sphere, unless otherwise stated. A subset of Rⁿ is called open if it contains a sphere of a positive radius about each of its points x. A subset is called closed if its complement is open.

(¶ f/¶ x)(x, y, z) º f_x(x, y, z) = lim_h® 0 [f(x+h, y, z)-f(x, y, z)]/h.

One could continue defining partial derivatives of partial derivatives which would be partial derivatives of higher order. Thus there are three first order partial derivatives of f(x, y, z) viz., f_x = ¶f/¶x, f_y = ¶f/¶y, and, f_z = ¶f/¶z, nine second order partial derivatives: f_xx º ¶²f/¶x² º ¶²f/¶x¶x º (¶/¶x)(¶/¶x)f, f_xy º ¶²f/¶x¶y º (¶/¶y) (¶/¶x)f º (¶/¶y) (¶f/¶x) = (f_x)_y, … , and so on.

f(a+h, b+k) - f(a, b) = Ah + Bk + o(|h| + |k|), as |h| + |k| ® 0.

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Theorem. If the first order partial derivatives f_x(x, y) and f_y(x, y) of the function f(x, y) exist in a neighborhood of a point (a, b) and are bounded there, then f(x, y) is continuous at (a, b).

Proof: The existence of partial derivatives in a neighborhood of (a, b) implies the continuity of f(x, y) on horizontal and vertical lines in a neighborhood of (a, b). Now, by the mean value theorem for functions of one variable, there exist q ₁, q ₂ Î (0, 1) such that f(x, y) – f(a, b) = [f(x, y) – f(a, y)] + [f(a, y) - f(a, b)] = f_x(a+q ₁(x-a), y)(x-a)+f_y(a, b+q ₂(y-b))(y-b) ® 0, as (x, y) ® (a, b). #

Chain Rule. If w = w(x) and x = x(t) then dw/dt = (dw/dx)(dx/dt) provided the right hand side is meaningful.

Proof: Note that w = w(x) = w(x(t)), Dx = Dx(t) = x(t+Dt)-D(x), etc. so that Dw/Dt = [w(x(t+Dt))-w(x(t))]/Dt =(Dw/Dx)(Dx/Dt). Passing to the limit as Dt ® 0, Dx ® 0 and we have the result. #

Chain Rule for Functions of Several Variables. Let w = w(x, y) and x = x(t), y = y(t). If (i) the partial derivatives ¶ w/¶ x, ¶ w/¶ y exist in a neighborhood of (x, y) = (x(t), y(t)) and are continuous at (x, y) and, (ii) x, y are differentiable at the point t, then

dw/dt = (¶w/¶x)(dx/dt) + (¶w/¶y)(dy/dt).

Dw/Dt = (w(x+Dx, y+Dy)–w(x, y))/Dt

= [(w(x+Dx, y+Dy)–w(x, y+Dy)/Dx][Dx/Dt] + [(w(x, y+Dy)–w(x, y)/Dy][Dy/Dt]

= [¶ w(x+q₁Dx, y+Dy)/¶x][Dx/Dt] + [¶w(x, y+q₂Dy)/¶x][Dy/Dt],

where 0 < q ₁, q ₂ <1. Passing to the limit as Dt ® 0, Dx, Dy ® 0, and the result follows. #

The chain rule obviously extends to a function f º f(x) º f(x₁, … , x_n) of n-variables n ³ 2, in which the partial derivatives are assumed to exist in a neighborhood of x = (x₁, … , x_n) and be continuous at the point x, with the vector variable x(t) being a differentiable functions of t:

df/dt = S _{1£ i£ n} (¶f/¶x_i)dx_i/dt.

Theorem (Interchange of Order of Partial Differentiation). Let the partial derivatives f_xy and f_yx of the function f(x, y) exist in a neighborhood of (a, b) and be continuous at the point (a, b). Then, f_xy(a, b) = f_yx(a, b).

Proof: Let g(x, y, a, b) = f(x, y) - f(x, b) – f(a, y) + f(a, b). By the mean value theorem used for the variable x, for some q₁ Î (0, 1),

g(x, y, a, b) = [f(x, y)-f(x, b)] – [f(a, y)-f(a, b)] = [f_x(a+q₁(x-a), y) - f_x(a+q₁(x-a), b)](x-a)

= f_xy(a+q₁(x-a), b+q₂(y-b))(x-a)(y-b),

for some q₂ Î (0, 1), by mean value theorem used for the variable y.

Next, by the mean value theorem used for the variable y, for some q₃ Î (0, 1),

g(x, y, a, b) = [f(x, y)-f(a, y)] – [f(x, b)-f(a, b)] = [f_y(x, b+q₃(y-b)) - f_y(a, b+q₃(y-b))](x-b)

= f_yx(a+q₄(x-a), b+q₃(y-b))(x-a)(y-b),

for some q₄ Î (0, 1), by mean value theorem used for the variable x.

f_xy(a, b) = lim_{(x, y)}® (a, b) g(x, y, a, b)[(x-a)(y-b)]^-1 = f_yx(a, b). #

Mean Value Theorem for Functions of Several Variables. Let f(x, y) be continuous on the line segment joining the points (a, b) and (a+h, b+k) and let the first order partial derivatives of f(x, y) exist in a neighborhood of the interior of the line segment joining (a, b) and (a+h, b+k) and be continuous at all interior points of the segment. Then, for some q Î (0, 1),

f(a+h, b+k) – f(a, b) = hf_x(a+qh, b+qk) + kf_y(a+qh, b+qk).

Proof: Define g(t) = f(a+th, b+tk). Since the partial derivatives f_x and f_y exist in a neighborhood of the interior of the line segment and are continuous at each point of the interior of the segment, the derivatives of g(t) at the interior points of the line segment could be evaluated by the chain rule. By the mean value theorem for functions of one variable for a q Î (0, 1), f(a+h, b+k)–f(a, b) = g(1)-g(0) = g¢(q ) = hf_x(a+qh, b+qk)+kf_y(a+qh, b+qk). #

Taylor’s Expansion with Remainder for Functions of Several Variables. Let f(x, y) be continuous on the line segment joining the points (a, b) and (a+h, b+k). Let the n-th order partial derivatives of f(x, y) exist in a neighborhood of the line segment joining (a, b) and (a+h, b+k) and be continuous at all points of the segment. Then, for some q Î (0, 1),

f(a+h, b+k) = f(a, b)+S_{1£ k£ n-1} (h¶/¶x+k¶/¶y)^jf(a, b)/j! + (h¶/¶x+k¶/¶y)ⁿf(a+qh, b+qk)/n!.

Proof: Note that the continuity of the n-th order partial derivatives at all points of the segment imply their boundedness in the neighborhood of each point of the segment, which in turn implies the continuity of the (n-1)-th and all lower order partial derivatives at all points of the segment. Hence, for the function g(t) = f(a+th, b+tk) by the chain rule for functions of several variables, g^(j)(t) = (h¶/¶x+k¶/¶y)^jf(a+th, b+tk), t Î [0, 1]. The result follows from the Taylor’s expansion g(1) = g(0) + S ₁£ j£ n-1 (j!)^-1g^(j)(0) + (n!)^-1g⁽ⁿ⁾(q), for some q Î (0, 1). #

Taylor’s Approximation for Functions of Several Variables. Let the n-th order partial derivatives of f(x, y) exist in a neighborhood of (a, b) and be continuous at (a, b). Then,

f(a+h, b+k) = f(a, b)+S _{1£ k£ n} (h¶/¶x+k¶/¶y)^jf(a, b)/j! + o(|h|+|k|)ⁿ, as (|h|+|k|) ® 0.

Proof: Let H = h/(|h|+|k|), K = k/(|h|+|k|), g(t) = f(a+tH, b+tK). As g¢ (0) = Hf_x(a, b) + Kf_y(a, b), g^(j)(0) = (H¶/¶x + K¶/¶y)^jf(a, b), 1 £ j £ n. By the Taylor’s expansion g(t) = g(0) + S_1£j£n g^(j)(0)t^j/j! + o(t)ⁿ, t ® 0. Hence, putting t = |h|+|k| in this relation we have f(a+h, b+k) = f(a, b) + S ₁£ j£ n (tH¶/¶x + tK¶/¶y)^jf(a, b)/j! + o(|h|+|k|)ⁿ = f(a, b) + S ₁£ j£ n (h¶/¶x + k¶/¶y)^jf(a, b)/j! + o(|h|+|k|)ⁿ, as (|h|+|k|) ® 0. #

 

Change of Variables in Double Integrals

Consider a parallelogram PQRS in the plane with the coordiantes P(x, y), Q(x+p, y+q), R(x+p+r, y+q+s), S(x+r, y+s). The vectors represented by the concurrent sides PQ and PS are v₁ = pi + qj and v₂ = ri + sj, respectively. If q is the angle between the vectors v₁ and v₂, |v₂| |sin q | is the hight of the parallelogram with |v₁| the base, so that

Area of Parallelogram = |v₁´ v₂| = |v₁||v₂| |sin q | = || = || = |ps-qr|.

Theorem (Change of variables in Double Integrals). Let D¢ be a domain in the uv-plane. Let x = x(u, v), y = y(u, v) be a smooth 1-1 map of D¢ onto a region D in the xy-plane. Then

ò ò _D f(x,y)dxdy = ò ò _D¢ f(x(u,v),y(u,v)|det J|dudv,

where the Jacobian J of the transformation is given by:J = ¶(x,y)/¶(u,v) = .

Proof: Let us use a rectangular mesh in D¢ . The rectangular area element DA¢ defined by the points {P¢(u, v), Q¢(u+Du, v), R¢(u+Du, v+Dv), S¢ (u, v+Dv)} in D¢ maps approximately into a parallelogram area D A defined by the points {P(x, y), Q(x + (¶x/¶u)Du, y + (¶y/¶u)Du), R(x + (¶x/¶u)Du + (¶x/¶v)Dv, y + (¶y/¶u)Du + (¶y/¶v)Dv), S(x + (¶x/¶v)Dv, y + ¶y/¶v)Dv)} in D. This is because of the first order approximations

x(u+Du, v+Dv) @ x(u, v) + (¶x/¶u)Du+(¶x/¶v)Dv,

y(u+Du, v+Dv) @ y(u, v) + (¶y/¶u)Du+(¶y/¶v)Dv,

for all sufficiently small Du, Dv. Hence the rectangular subdivision in D¢ induces a subdivision in D consisting of approximate parallelograms.

Note that DA¢ = DuDv and DA = |det J|DuDv. Hence the Riemann sum

S f(x(u, v), y(u, v)) |det J|DuDv ® òò_D f(x, y))dxdy.

But this Riemann sum is also an approximation to the integral ò ò_D¢ f(x(u, v), y(u, v)) |det J| dudv. Hence, we must have

òò _D f(x, y))dxdy = ò ò_D¢ f(x(u, v), y(u, v)) |det J| dudv.

Example. For the cartesian to polar transformation x = r cos q , y = r sin q , J = ¶(x,y)/¶(r,q) = , |det J| = r, and therefore if a region D¢ in the (r, q ) plane is 1-1 mapped onto the region D in the (x, y) plane,

òò _D f(x,y)dxdy = òò _D¢ f(r cos q, r sin q) rdrdq . (Polar coordinates)

Thus, the area of a circle D of radius R with center at (0, 0) in the xy-plane is mapped on to the rectangle D¢ = [0, R]´[0, 2p ] in the rq -plane. Accordingly the area of the circle òò_D dxdy = òò_D¢ rdrdq = ò_{(0,2p )} [ò_(0,R) rdr]dq = (R²/2)2p = pR². Note that the evaluation òò_D dxdy = ò_(-R,R) [ò_{(-Ö (R2-x2),Ö (R2-x2))} dy]dx = ò_(-1,1) 2Ö(R²-x²)dx = (putting x = R sin q ) ò_{(-p/2,p /2)} 2Rcos²q Rdq = R² ò_(-p/2,p/2) [1 + cos 2q ] dq = R²[q + (sin 2q)/2]| _(-p/2,p/2) = pR², is much more complicated.

Volume of a Parallelopiped

Volume of Parallelopiped = |[v₁, v₂, v₃]| = |(v₁´v₂)×v₃| = |det|.

Note that v₁´v₂ relates to the base area, and the subsequent dot product multiplies the base area with the hight of the parallelogram along with a sign which is positive if the angle between v₁´ v₂ is acute, negative if the angle is obtuse. If v₃ lies in the plane of v₁, v₂ the area, indeed, is zero.

Theorem (Change of variables in Triple Integrals). Let the change of variables

x = x(u, v, w), y = y(u, v, w) , z = z(u, v, w)

map a region D¢ in the uvw-space 1-1 onto a region D in the xyz-space. Then,

òò_D f(x, y)dxdydz = òò_D¢ f(x(u, v), y(u, v)|det J|dudvdw,

where the Jacobian J of the transformation is given by:

J = ¶(x,y,z)/¶(u,v,w) = .

Proof: Consider a subdivision of D¢ through rectangular parallelopipeds with edges pararallel to to coordinate axes in the uvw-space. Let the coordinates of the vertices O, P, Q, R associated with the concurrent edges OP, OQ, OR of a rectangular parallelopiped volume element be (u, v, w), (u+Du, v, w), (u, v+Dv, w), (u, v, w+Dw). The change of variable maps this volume element into an approximate parallelopiped with the associated concurrent edge vectors as: v₁ = (¶x/¶u)Du i + (¶y/¶u)Du j + (¶z/¶u)Du k, v₂ = (¶x/¶v)Dv i + (¶y/¶v)Dv j + (¶z/¶v)Dv k, v₃ = (¶x/¶w)Dw i + (¶y/¶w)Dw j + (¶z/¶w)Dw k, which therefore has approximate volume as DV = |det J|DuDvDw. Thus the Riemann sum S f(x, y, z) |det J|DuDvDw ® òòò_D f(x, y, z)dxdydz. As the sum is also a Riemann sum in R¢ associated with the integral òòò_D¢ f(x(u, v, w), y(u, v, w), z(u, v, w))|det J|dudvdw, by the convergence of the Riemann sums to the associated integrals, we have òòò_D f(x, y, z)dxdydz = òòò_D¢ f(x(u, v, w), y(u, v, w), z(u, v, w)) |det J| dudvdw, which is the required change of variable formula. #

Example. The transformation from the cartesian coordinates (x, y, z) to spherical coordinates (r, q, f) is given by: x(r, q, f) = r cos q sin f, y(r, q , f) = r sin q sin f , z(r, q , f) = r cos f , r Î [0, ¥), q Î [0, 2p), f Î [0, p ]. The Jacobian of the transformation is

so that, det J = r² [cosf (-sin²q sinf cosf - cos²q sinf cosf) – sinf (cos²q sin²f + sin²q sin²f)] = - r² sin f .

The negative sign signifies that the infinitesimal vector triad forms a left handed system. Since, |det J|= r² sin f,

òòò_R f(x,y,z)dxdydz = òòò_R¢ f(rcosqsinf, rsinqsinf, rcosf) r²sinfdrdqdf, (Spherical)

where the transformation maps the region R¢ in the sphereical system to the region R in the cartesian system.

The transformation from the cartesian (x, y, z) to cylindrical coordinates (r, q, z) consists of x(r, q) = x(r, q, z) = r cos q, y(r, q) = y(r, q, z) = r sin q, z(r, q, z) = z, in which case we similarly have det J = r, so that

òòò_R f(x, y, z)dxdydz = òòò_R¢ f(rcosq, rsinq, z) rdrdqdz. (Cylindrical)

Let f = f(x, y, z) º f(xi + yj + zk) º f(P) , be a scalar function. Its directional derivative at the point P = P(x, y, z) in the direction of a vector b (written as ¶f/¶s, or D_bf, or ¶ f/¶ u, where u = b/|b|) is defined by

¶f/¶s = lim_{s® 0} [f((Q)-f(P)]/s

¶f/¶s = lim_s®0 [f(v+su)–f(v)]/s = lim_s®0 [f(v+su)–f(v)]/s = (¶f/¶x)u₁ + (¶f/¶y)u₂ + (¶f/¶z)u₃, = Ñf×u = Ñf×(b/|b|),

where Ñ , the gradient operator is given by

grad = Ñ = i¶/¶x + j¶/¶y + k¶/¶z,

grad f º Ñf = (¶f/¶x)i + (¶f/¶y)j + (¶f/¶z)k.

¶f/¶s = |Ñf| cos g ,

where cos g denotes the direction cosine between Ñf and b, i.e., g is the angle between the gradient and the direction vector b, the directional derivative is maximum in the direction of the gradient and the maximum value is the magnitude of the gradient. Since of the maximum rate of change and its direction depends on the function only the gradient Ñ f is independent of the choice of the Cartesian coordinate system in the space, i.e., the origin of the coordinate system and the choice of the orthogonal x, y, z directions.

R = (-¥, ¥) - the real line.

R³ = {(x, y, z) : x, y, z Î R} - the 3-D space.

The operator Ñ

Ñ = i¶/¶x + j¶/¶y + k¶/¶y

The operator Ñ is called the operator del.

f º f(x,y,z) - a scalar function in R ³ of the variables x, y, z.

grad f º Ñf = (i¶/¶x + j¶/¶y + k¶/¶y)f(x, y, z) = i¶f/¶x + j¶f/¶y + k¶f/¶z.

The gradient Ñf of f(x, y, z) is a vector.

F(x, y, z) º if₁(x, y, z) + jf₂(x, y, z) +kf₃(x, y, z) - a vector function in R³ of the variables x, y, z.

div F º Ñ×F = (i¶/¶x + j¶/¶y + k¶/¶y)× (if₁ + jf₂ +kf₃) = ¶f₁/¶x + ¶f₂/¶y + ¶f₃/¶z.

The divergence Ñ×F of F(x, y, z) is a scalar function of x, y, z.

F(x, y, z) º if₁(x, y, z) + jf₂(x, y, z) + kf₃(x, y, z) - a vector function in R³ of the variables x, y, z.

curl F º Ñ ´ F = (i¶/¶x + j¶/¶y + k¶/¶z)´(if₁ + jf₂ +kf₃) = i(¶f₃/¶y - ¶f₂/¶z) + j(¶f₁/¶z - ¶f₃/¶x) + k(¶f₂/¶x - ¶f₁/¶y)

Let C = {r(t) : a £ t £ b} be a rectifiable space curve, i.e., S ₁£ i£ n |r(t_i) - r(t_i-1)| ® L, n ® ¥ , irrespectively of the subdivision a = t₀ < t₁ < … < t_n = b, so long as the maximum subdivision length max₀£ i£ n | t_i - t_i-1| ® 0. The line integral ò _C fds, (s denoting the arc length along C, and f º f(r(t)) º f(x(t), y(t), z(t)) º f(x, y, z)), is defined (in the Riemann sense) as

ò_C fds = lim_n®¥ S_{1£ i£ n} f(r(t_i))|r(t_i) - r(t_i-1)|,

where t_i-1 £ t_i £ t_i is an arbitrary point, provided limits of all such sums exist and are equal. The line integral of the function [f(x, y, z)dx/ds + g(x, y, z)dy/ds + h(x, y, z)dz/ds], is simply written as ò_C (fdx + gdy + hdz). Note that ò_C fds = ò_(a,b) f(x(t), y(t), z(t))|r¢(t)|dt = ò_(a,b) f(x(t), y(t), z(t))(ds/dt)dt = ò_(0,L) f(x(s), y(s), z(s))ds, and

ò_C (fdx+gdy+hdz) = ò_(0,L) (fdx/dt+gdy/dt+hdz/dt)(dt/ds)ds = ò_(a,b) (fdx/dt+gdy/dt+hdz/dt)dt.

If v is a vector function, the integral ò_C v×dr is defined as

ò_C v×dr = ò_C (v₁dx + v₂dy + v₃dz).

Let S denote a surface and DA₁, DA₂, … , DA_n denote the element areas of a subdivision of the surface into small area elements. Let P_i º (x_i, y_i, z_i) denote an arbitrary point on the area element DA_i. The surface integral ò_S gdA is defined (in the Riemann sense) as

ò_S gdA = lim_DA®0 S_{1£ i£ n} g(x_i, y_i, z_i) DA_i,

where DA = max_i diameter (DA_i) ® 0 (the diameter of a set being the diameter of the samllest circle (in 3-D sphere) containing the set), the limit being independ of the choice of subdivision obtained by drawing a piecewise smooth mesh on S.

Let a surface be described by a two parameter vector r = r(u, v), (u, v) Î D¢, D¢ being a planar region and wite g(r) as g(u, v). The area of the approximate parallelogram points on S associated with the points (u, v), (u+Du, v) (u, v+Dv) (u+Du, v+Dv) in D¢ being r, r+r_uDu, r+r_vDv, r+r_uDu+r_vDv, is given by |r_u´r_v|DuDv. It follows that

òò _S gdA = òò_D¢ g(u, v) |r_u´r_v|dudv.

If the surface S is described as z = f(x,y), (x, y) Î D, i.e., r º r(x, y) = ix + jy + kf(x, y), |r_x´r_y| = |(i + kf_x)´(j + kf_y)| = |k – jf_y – if_x| = Ö(1 + f_x² + f_y²), so that writing g º g(x, y, f(x, y))

òò _S gdA = òò_D g(x, y, f(x, y))Ö(1 + f_x² + f_y²) dxdy.

An oriented surface is a surface with a continuous principal normal n at each of its points. The boundary curve of such a surface is given a positive sense of traversal when it follows the fingers while the normal points in the direction of the thumb of the right hand. For an oriented surface z = f(x, y), any tangential element (idx + jdy + kdz) satisfies dz – f_xdx – f_ydy = 0, so that the choice of an upward normal direction is given by – if_x – jf_y + k, with the principal normal being n = (– if_x – jf_y + k)/Ö(1 + f_x² + f_y²). If v = v(x, y, z) is a vector function defined at points of S, we use the notation v_n = (v)_n º v×n. Hence,

òò_S (v)_ndA = òò_D (-v₁f_x –v₂f_y + v₃)/Ö(1 + f_x² + f_y²) dA = òò_D (– v₁f_x – v₂f_y + v₃)dxdy.

The integral òò_S (v)_n dA is sometimes also denoted as òò_S v×dA, where ‘dA’ denotes the vector area element ‘ndA’. Writing n = cos a i + cos b j + cos g k, thus cosa , cos b , cos g denoting the direction cosines of the normal to the surface at a point, thus we have: òò_S (v)_ndA = òò_S v× ndA = òò_S v×dA = òò_S (v₁ cos a + v₂ cos b + v₃ cos g ) dA = òò_S (v₁ dydz + v₂ dzdx + v₃ dxdy). Note that neglecting higher order terms "D xD y" is the projection "cos g D A" of the area element "D A" on the surface associated with the xy-plane rectangle defined by the points (x, y), (x+D x, y), (x, y+D y), (x+D x, y+D y), and similarly for "D yD z" and "D zD x".

A volume integral òòò_R f dV is defined as

òòò_R f dV = lim_DV®0 S_{1£ i£ n} f(x_i, y_i, z_i) DV_i,

in the Riemann sense, where the element volumes DV_i, 1 £ i £ n, constitute a subdivision of the volume region R into small volume elements, DV = max_i diameter (DV_i), provided the limit is independent of the choice of subdivision. It is clear that the volume integral òòò_R f dV is the same as the triple integral òòò_R f(x, y, z)dxdydz.

Let R be a closed bounded region in the xy-plane whose boundary C consists of finitely many smooth curves. Let f(x, y) and g(x, y) be functions which are continuous and have continuous partial derivatives ¶ f/¶ y and ¶ g/¶ x everywhere in some domain containing R. Then

.

Proof: Consider R described by {a £ x £ b, p(x) £ y £ q(x)}, as well by {c £ y £ d, r(y) £ x £ s(y)}. Then,

òò_R (¶f/¶y)dxdy = ò_(a,b) [ò_(p(x),q(x) (¶f/¶y)dy]dx = ò_(a,b) [f(x, q(x)) - f(x, p(x))]dx = - ò_C fdx,

òò_R (¶g/¶x)dxdy = ò_(c,d) [ò_(r(y),s(y) (¶g/¶x)dx]dy = ò_(c,d) [g(s(y), y) - g(r(y), y)]dx = ò_C gdy.

Combining we have òò_R (¶g/¶x - ¶f/¶y)dxdy = ò_C (fdx+gdy). The result for a general region is proved by breaking it into a union of such regions and noting that for a common boundary of two such regions the two line integrals, being negatives of each other, cancel. #

Let R be a closed bounded region in space with a piecewise smooth orientable boundary surface S. Let u(x, y, z) be a vector function which is continuous and has continuous first partial derivatives in some domain containing R. Let n denote the unit outer normal and u_n = u×n the normal component of u on the boundary S of R. Then

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Proof: Consider R described by {z₁(x, y) £ z £ z₂(x, y) : (x, y) Î D}, where z₁, z₂ describe the surfaces S₁ and S₂ with the upward principal normals n₁ and n₂. Then, writing n = cos a i + cos b j + cos g k,

òòò_R (¶u₃/¶z)dV = òò_D [u₃(x, y, z₂) - u₃(x, y, z₂))dxdy = òò_S2 u₃ cos g dA - òò_S1 u₃ (-cos g )dA = òò_S u₃ cos g dA.

Adding to the above the corresponding expressions for òòò_R (¶u₁/¶z)dV, and òòò_R (¶u₂/¶z)dV, assuming R to be described similarly with respect to x and y also, we get

òòò_R Ñ×u dV = òòò_R (¶u₁/¶z + ¶u₂/¶z + ¶u₃/¶z)dV = òò_S (u₁ cos a + u₂ cos b + u₃ cos g) dA = òò_S u×n dA,

Let S be a piecewise smooth oriented surface in space and let the boundary of S be piecewise smooth simple closed curve C. Let v(x, y, z) be a continuous vector function which has continuous first partial derivatives in a domain in space which contains S. Let (curl v)_n = (curl v)×n denote the normal component of curl v on the surface S, and v_t = v×t the tangential component of v with respect to the boundary C of S. Then

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Proof: By breaking the surface into a union of pieces the integrals along the common boundaries cancel and the result follows from that on the individual pieces. This enables a convenient description of each piece. Hence without loss of generality let S be such a piece and C its boundary. Let n = i cos a + j cos b + k cos g . Then

òò_S (Ñ´v)_ndA = òò_S [(¶v₂/¶x-¶v₁/¶y)cosg +(¶v₃/¶y-¶v₂/¶z)cosa +(¶v₁/¶z-¶v₃/¶x)cosb ]dA.

The contribution of the terms in v₁ on the right hand side equals òò_S [(-¶v₁/¶y)cos g + (¶v₁/¶z)cos b )] A. If S lies in a plane parallel to the yz-plane, the contributions of v₁ terms are zero both in the surface and the boundary integrals. Hence without loss of generality we could describe S by z = f(x, y), (x, y) Î D (else the roles of z and y could be interchanged). Then n = (-f_x i -f_y j + k)/Ö(f_x² + f_y² + 1) and the surface integral evaluated as a double integral gives

òò_S [(-¶v₁/¶y)cos g + (¶v₁/¶z)cos b )]dA = - òò_D [(¶v₁/¶y) + (¶v₁/¶z)f_y]dxdy = - òò_D [¶v₁(x, y, f(x, y))/¶y]dxdy

= ò_Bd(D) v₁(x, y, f(x, y))dx = = ò_Bd(D) v₁(x, y, f(x, y))(dx/ds)ds,

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