Let f function from (R,d) to (R,d ) defined as f(x) = x , where d is .the discrete metric then f is continuos True False

Q: (a) Find the points (x ,y, z) in space, where the function f(x, y, z) = Vx² +y -1 Is continuous.

A: It is well known that the equation z=x2+y2 represents a circle with centre at the origin. The…

Q: (a) Assume that g(x, y) is a function of the form g(x, y) = 0 if (x, y) = (0,0). (i) Show that g(x,…

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering the…

Q: Verify divergence theorem for f=x2i+zi+yzk taken over the cube bounded by x=0,x=1,y=0,y=1,z=0 and…

A: Use the divergence theorem to solve this problem.

Q: Show that each function in satisfies a Laplaceequation. ƒ(x, y, z) = 2z3 - 3(x2 + y2)z

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Q: Let f(z) = u(x, y)+iv(x,y) be entire such that u(x, y) 4v(x, y). Show that f is constant. %3D

A: Here we use CRE equation to prove this

Q: Please show step-by-step

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Q: Let f (x, y) = g(u), where u = x² + y² and g(u) is differentiable. Prove that () = 4u du + ax 2.

A: Given, f(x,y)=g(u)=x2+y2 To prove that, ∂f∂x2+∂f∂y2=4udgdu2

Q: (a) Assume that g(x, y) is a function of the form g(x, y) 0 if (x, y) = (0,0). (i) Show that g(x,…

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Q: H.W where or is the selid bounded Sur faces x*+g=¿ , the

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Q: sume that f(t) is continuous on [a, b| and that u = g(t) is cont ferentiable on [a, b], then f(u)…

A: Note- Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Define h(0,0) in such a way that h is continuous at origin       h(u,v)=ln(nu2-u2v2+nv2/u2+v2

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Q: Let F(x, y) =x2+xyz+z , find at the point (1,1,1)

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Q: f(z) = Im z is continuous 0 صواب ihi O

A: True

Q: Prove that the function f : R? → R, f(x, y) = sin(/x² +|sin y|) is uniformly continuous on R?.

A: The time interval between two waves is known as a Period whereas a function that repeats its values…

Q: \(d\) is a metric on \(X\). Show that \(\rho : X^2 \to [0,\infty)\) defined by \[ \rho(x,y) =…

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Q: Define h(0,0) in such a way that h is continuous at origin nu? – u?v² + nv² u? + v? h(u, v) = In

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Q: If a vector-valued function r(t) is continuous at t = a, then the limit r(a+ h) – r(a) lim exist.…

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Q: Define f: R² → R by sin(ry) f(x,y) := TY if ry # 0, f(t,0) = 1 = f(0, t) Vt e R. Prove that f is…

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Use integration by parts to evaluate lim Sest f(1)dt. (Let u = Z→∞J f(t) and dv = est dt.)

A: As per the question we are given a limit of an integral, and we have to use the integration by parts…

Q: (a) Assume that f(x, y, z) and F (x, y, z) are arbitrary differentiable functions such that f(x, y,…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: If u(z,0) = f(z) ue(z,0) = 0 u(z, t), u(z,t) → 0 as |z| → 0 Show that f(z+at) + f(z- at) u(z,t) =…

A: If we have a D'Alembret wave equation given by utt=α2uxx       , -∞<x<∞,   t>0, α is fixed…

Q: c) Let f(r, y) = Vlry| Prove that f(r, y) is not differentiable at point (0, 0).

A: Given a function : f(x,y)=xy We need to show that f(x ,y) is not differentiable at (0 ,0). Let's…

Q: At what points (x,y.z) in space are the tunctions continuous? a. h(x.y.z) = In (4z² - 4x – 2y³ - 5)…

A: This is a problem of multi variable calculus.

Q: Suppose that F(x,y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that дх…

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Q: Let f(x+ y) = f(x).s) Vx, ye R.Prove that f is derivable on R iff it is derivable at one point of R.

A: If the given function is continuous at x=a then it is continuous at x=b.

Q: -dt. dF = (dr · V)F+ at

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Q: Let u be a differentiable function of x. Use the fact that ∣u∣ = √u2 to prove that

A: Given that - u = u2          ............(1)To prove - ddxu = u'uuProof - Difefrentiating equation…

Q: . Show that the function u = p(ry) + Vry v(y/x) provides the quation = 0. ду? %3D dx²

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Q: Let f(x, y) = x³ + 2ry – y² be a func the tangent plane to f(x, y) at the poi

A: Here we have f(x,y)=x3+2xy-y2  we have to find the equation of the tangent plane to f(x,y) at point…

Q: If f(x, y) is continuous on [a, b] × [c, d], and g(x, y) = Sª S' f(s,t)dsdt, prove that, for a < x <…

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Q: Let S' denote the unit circle in R². Integrate the function f(x, y) = x² + y² over the unit circle…

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Q: i) True or False: T = {0, R}U {(-x, a] : a € R} is a topology on R.

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Q: Suppose f(x, y) is continuous and non-negative over a non-empty open region R in the plane with a…

A: Based on given data,For Non empty open region R,fx, y is continuous,andfx, y is non -…

Q: /Evaluate The are bounded by The Semiellipes y=25-x' oP The lines X = ±1,Y=-1

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Q: If f(z) = u + iv is entire and u(x, y) is bounded above, apply Liouville's Theorem to exp(f(z)) to…

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Q: Use Green's theorem to compute the circulation of F(x, y) = (-y, x²) around the perimeter of the…

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Q: Use the Divergence Theorem to evaluate J] F. ds where F (x, y, =) = , and S is the unit S ball x +…

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Q: ƏF dF = (dr. V)F+ -dt. at

A: Given: A vector function F depends on both space coordinates x, y, z and time t.  To Find:  To show…

Q: Define f : [0, ∞) → [0, ∞) by f(x) = Vx. (a) (b) Prove that va +b 0. Prove that f is uniformly…

A: We have to prove that   √a+√b≤ √(a + b) for all a, b ≥ 0.

Q: ) Let z(x, y) = x" be a function defined on a disk D in the positive quadrant the point (1, 2).…

A: As per the question we are given a bivariate function : z(x,y) = xy Defined on a disc D containing…

Q: Let r be a di¡erentiable vector function of t. Show that if r # (dr/dt) = 0 for all t, then  r  is…

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Q: Let f : C[0,1] → P be given byf (x ) = x (1)Vx e C[0,1]. Show that f is continuous w.r.t the supnorm…

A: f is continuous with respect to sup norm. Let xn be a sequence of continuous functions in 0,1 such…

Q: d(r, y) 1+d(x,y) 5. Let (X, d) be a metric space. Define f : X x X → R by f(r, y) = Show that f is a…

A: (χ , d)  be a metric space , define f :χ×χ→ℝ by f(x , y)=d(x , y)1+d(x , y) Positive definite…

Q: R be cont g is dlferentlable and compute g. ppose f: [a, b] → R is continuous and g : [c, d] → [a,…

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Q: Consider the vector space of continuous functions over the interval [-1,1] with the inner product…

A: Given that,

Q: Find the limit of the vector-valued function if it exists. lim(e² i + sin(t)j + e¯¹²k) O does not…

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Q: Study the continuity of f(x,y) = xe-Vx at p(1,0) ey.

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