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8.1).We shall show that if f (x) is an invertible linear mapping, then we also haveh = O(t) as t → 0.Indeed, we find successively by (8.51) that) *−1)*−1f (x) t = h + f (x) o(h)) *−1f (x) t = h + o(h)===) *−1 == f (x) t = ≥ ,h, − =o(h)==) *−1 = 1= f (x) t = ≥ ,h,2as h → 0,as h → 0,as h → 0,(8.52)for ,h, < δ,where the number δ > 0 is chosen so that ,o(h), < 12 ,h, when ,h, < δ. Then,taking account of (8.50), that is, the relation h → 0 as t → 0, we find=) *−1 =,h, ≤ 2= f (x) t = = O ,t, as t → 0,4488 Differential Calculus in Several Variableswhich is equivalent toh = O(t)as t → 0.From this it follows in particular thato(h) = o(t) as t → 0.Taking this relation into account, we find by (8.50) and (8.52) that*−1)h = f (x) t + o(t) as t → 0or)*−1f −1 (y + t) − f −1 (y) = f (x) t + o(t)as t → 0.It is known from algebra that if the matrix A corresponds to the linear transformation L : Rm → Rm , then the matrix A−1 inverse to A corresponds to the linear transformation L−1 : Rm → Rm inverse to L.
The construction of the elementsof the inverse matrix is also known from algebra. Consequently, the theorem justproved provides a direct recipe for constructing the mapping (f −1 ) (y).We remark that when m = 1, that is, when Rm = R, the Jacobian of the mappingf : U (x) → V (y) at the point x reduces to the single number f (x) – the derivative of the function f at x – and the linear transformation f (x) : T Rx → T Ryreduces to multiplication by that number: h → f (x)h.
This linear transformation is invertible if and only if f (x) = 0, and the matrix of the inverse mapping[f (x)]−1 : T Ry → T Rx also consists of a single number, equal to [f (x)]−1 , thatis, the reciprocal of f (x). Hence Theorem 4 also subsumes the rule for finding thederivative of an inverse function proved earlier.8.3.4 Problems and Exercises1. a) We shall regard two paths t → x1 (t) and t → x2 (t) as equivalent at the pointx0 ∈ Rm if x1 (0) = x2 (0) = x0 and d(x1 (t), x2 (t)) = o(t) as t → 0.Verify that this relation is an equivalence relation, that is, it is reflexive, symmetric, and transitive.b) Verify that there is a one-to-one correspondence between vectors v ∈ T Rmx0and equivalence classes of smooth paths at the point x0 .c) By identifying the tangent space T Rmx0 with the set of equivalence classes ofsmooth paths at the point x0 ∈ Rm , introduce the operations of addition and multiplication by a scalar for equivalence classes of paths.d) Determine whether the operations you have introduced depend on the coordinate system used in Rm .8.3 The Basic Laws of Differentiation4492.
a) Draw the graph of the function z = x 2 + 4y 2 , where (x, y, z) are Cartesiancoordinates in R3 .b) Let f : G → R be a numerically valued function defined on a domain G ⊂Rm . A level set (c-level) of the function is a set E ⊂ G on which the functionassumes only one value (f (E) = c).
More precisely, E = f −1 (c). Draw the levelsets in R2 for the function given in part a).c) Find the gradient of the function f (x, y) = x 2 + 4y 2 , and verify that at anypoint (x, y) the vector grad f is orthogonal to the level curve of the function fpassing through the point.d) Using the results of a), b), and c), lay out what appears to be the shortest pathon the surface z = x 2 + 4y 2 descending from the point (2, 1, 8) to the lowest pointon the surface (0, 0, 0).e) What algorithm, suitable for implementation on a computer, would you propose for finding the minimum of the function f (x, y) = x 2 + 4y 2 ?3. We say that a vector field is defined in a domain G of Rm if a vector v(x) ∈ T Rmxis assigned to each point x ∈ G.
A vector field v(x) in G is called a potential fieldif there is a numerical-valued function U : G → R such that v(x) = grad U (x). Thefunction U (x) is called the potential of the field v(x). (In physics it is the function−U (x) that is usually called the potential, and the function U (x) is called the forcefunction when a field of force is being discussed.)a) On a plane with Cartesian coordinates (x, y) draw the field grad f (x, y)for each of the following functions: f1 (x, y) = x 2 + y 2 ; f2 (x, y) = −(x 2 + y 2 );f3 (x, y) = arctan(x/y) in the domain y > 0; f4 (x, y) = xy.b) By Newton’s law a particle of mass m at the point 0 ∈ R3 attracts a particle ofmass 1 at the point x ∈ R3 (x = 0) with force F = −m|r|−3 r, where r is the vector−→Ox (we have omitted the dimensional constant G0 ).
Show that the vector field F(x)in R3 \0 is a potential field.c) Verify that masses mi (i = 1, . . . , n) located at the points (ξi , ηi , ζi ) (i =1, . . . , n) respectively, create a Newtonian force field except at these points and thatthe potential is the functionU (x, y, z) =ni=1#mi(x − ξi )2 + (y − ηi )2 + (z − ζi )2.d) Find the potential of the electrostatic field created by point charges qi (i =1, . . .
, n) located at the points (ξi , ηi , ζi ) (i = 1, . . . , n) respectively.4. Consider the motion of an ideal incompressible liquid in a space free of externalforces (in particular, free of gravitational forces).Let v = v(x, y, z, t), a = a(x, y, z, t), ρ = ρ(x, y, z, t), and p = p(x, y, z, t) berespectively the velocity, acceleration, density, and pressure of the fluid at the point(x, y, z) of the medium at time t.An ideal liquid is one in which the pressure is the same in all directions at eachpoint.4508 Differential Calculus in Several Variablesa) Distinguish a volume of the liquid in the form of a small parallelepiped, oneof whose edges is parallel to the vector grad p(x, y, z, t) (where grad p is takenwith respect to the spatial coordinates).
Estimate the force acting on this volumedue to the pressure drop, and give an approximate formula for the acceleration ofthe volume, assuming the fluid is incompressible.b) Determine whether the result you obtained in a) is consistent with Euler’sequationρa = − grad p.c) A curve whose tangent at each point has the direction of the velocity vectorat that point is called a streamline. The motion is called stationary if the functionsv, a, ρ, and p are independent of t.
Using b), show that along a streamline in thestationary flow of an incompressible liquid the quantity 12 ,v,2 + p/ρ is constant(Bernoulli’s law5 ).d) How do the formulas in a) and b) change if the motion takes place in thegravitational field near the surface of the earth? Show that in this caseρa = − grad(gz + p).so that now the quantity 12 ,v,2 + gz + p/ρ is constant along each streamline of thestationary motion of an incompressible liquid, where g is the gravitational acceleration and z is the height of the streamline measured from some zero level.e) Explain, on the basis of the preceding results, why a load-bearing wing has acharacteristic convex-upward profile.f) An incompressible ideal liquid of density ρ was used to fill a cylindrical glasswith a circular base of radius R to a depth h.
The glass was then revolved about itsaxis with angular velocity ω. Using the incompressibility of the liquid, find the equation z = f (x, y) of its surface in stationary mode (see also Problem 3 of Sect. 5.1).g) From the equation z = f (x, y) found in part f) for the surface, write a formulap = p(x, y, z) for the pressure at each point (x, y, z) of the volume filled by therotating liquid. Check to see whether the equation ρa = − grad(gz + p) of part d)holds for the formula that you found.h) Can you now explain why tea leaves sink (although not very rapidly!) andwhy they accumulate at the center of the bottom of the cup, rather than its side,when the tea is stirred?5. Estimating the errors in computing the values of a function.a) Using the definition of a differentiable function and the approximate equalityΔf (x; h) ≈ df (x) h, show that the relative error δ = δ(f (x); h) in the value ofthe product f (x) = x 1 · · · x m of m nonzero factorsdue to errors in determining thefactors themselves can be found in the form δ ≈ mi=1 δi , where δi is the relativeerror in the determination of the ith factor.5 Daniel Bernoulli (1700–1782) – Swiss scholar, one of the outstanding physicists and mathematicians of his time.8.3 The Basic Laws of Differentiation4511b) Using the equality d ln f (x) = f (x)df (x), obtain the result of part a) againand show that in general the relative error in a fractionf1 · · · fn(x1 , .
. . , xm )g1 · · · gkcan be found as the sum of the relative errors of the values of the functionsf 1 , . . . , f n , g 1 , . . . , gk .6. Homogeneous functions and Euler’s identity. A function f : G → R definedin some domain G ⊂ Rm is called homogeneous (resp. positive-homogeneous) ofdegree n if the equalityf (λx) = λn f (x) resp.
f (λx) = |λ|n f (x)holds for any x ∈ Rm and λ ∈ R such that x ∈ G and λx ∈ G.A function is locally homogeneous of degree n in the domain G if it is a homogeneous function of degree n in some neighborhood of each point of G.a) Prove that in a convex domain every locally homogeneous function is homogeneous.b) Let G be the plane R2 with the ray L = {(x, y) ∈ R2 | x = 2 ∧ y ≥ 0} removed. Verify that the function!y 4 /x, if x > 2 ∧ y > 0,f (x, y) =at other points of the domain,y3,is locally homogeneous in G, but is not a homogeneous function in that domain.c) Determine the degree of homogeneity or positive homogeneity of the following functions with their natural domains of definition:f1 x 1 , . .
. , x m = x 1 x 2 + x 2 x 3 + · · · + x m−1 x m ;f2 x 1 , x 2 , x 3 , x 4 =x1x2 + x3x4;+ x2x3x4l f3 x 1 , . . . , x m = x 1 · · · x m .x1x2x3d) By differentiating the equality f (tx) = t n f (x) with respect to t, show that ifa differentiable function f : G → R is locally homogeneous of degree n in a domainG ⊂ Rm , it satisfies the following Euler identity for homogeneous functions:x1 1 1∂f 1mm ∂fmmx+···+xx≡nfx.,...,x,...,x,...,x∂x m∂x 1e) Show that if Euler’s identity holds for a differentiable function f : G → R ina domain G, then that function is locally homogeneous of degree n in G.Hint: Verify that the function ϕ(t) = t −n f (tx) is defined for every x ∈ G and isconstant in some neighborhood of 1.4528 Differential Calculus in Several Variables7.
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