Let d be metric on X, and de f ined di on X as di = max (d (r, ).0}, then dis semi metric but not metric C not well defined O

Q: the function is a metric of R: d(x,y) = sqrt(x^2−y^2);  x,y ∈ R

A: let z1=(x1,y1) and z2=(x2,y2) now, d(X x Y, d) is a metric space if 1)d(z1,z2)=0 if z1=z2 now, if…

Q: Find Equatien of the tavgnt lin tu th cuve at tk giver pointe y= sec X ,

A:

Q: Please help me number 6. Thanks!

A: We are given that the unit vectors x, y, z satisfies the equationTo find the relation between the…

Q: 4. For the cylinder with patch X(u, v) = (R cos u, R sin u, v), (a) Find the normal vector U. (b)…

A: Given Data: Let us consider the given data, Xu,v=Rcosu,Rsinu,v. A non-zero vector that is…

Q: 2. Find the tangent plane to z = x f (t) where t = at a point (a,b) not on the y-axis. [Note: be…

A:

Q: E valuate f win Tydy tyxdx olE Cethe line seqment f&Om (0,2) to

A: on solving this we get the

Q: 4. Does there exist a simple closed unit-speed plane curve with curvature less than or equal to 2…

A: In a simple closed unit-speed plane curve, curvature =1r According to the question, Area of the…

Q: I. (a + y?) da dy dz; 2: 2<4-a? - ở, z 20 (x².

A:

Q: function flx) is defined by fa)= tind fby wsing the मिर वक Theoremor Fundamental Calculus.

A:

Q: F = e* ((1+xy), x²) and C:r(t) = (cost, 2sin t) Conservation, Independence Path a) Determine whether…

A:

Q: Consider the directional derivative of the general function z =, = f(x,y). a) Where is it a i)…

A: The directional derivative of function φ=z-f(x,y) at (a,b,c) in the direction of u…

Q: 9. GIVEN: QU || DA, QU DA. PROVE: AUXQ ADXA.

A:

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

A:

Q: Show that the tangent at P(x,y,) the curve Vx+ Vy = Va is yy,² +xx, ² = a²

A:

Q: If N is a sphere of radius R that is centered around the origin. Use appropriate functions and…

A: Simplify the integral:  ∫Ωr→r3∇→r4-R4r5+7r3+2dτ=∫Ωrr3r^r^ddrr4-R4r5+7r3+2r2sinθdrdθdϕ…

Q: The curvature of a point moving along the curve given as R(t)=(In sec t) i+ tj is: O t O sec t O…

A: Here, Rt=lnsecti+tj

Q: 4. Let r(s) be a unit speed curve with a curvature of zeroc fo. prove that the image r lies in a…

A:

Q: 3) use implict differentiation to finol dd et the point (8.1). 4 xy+ slny=32 Anywery: 28 a) 37 り 36…

A:

Q: 5. Given r = F(0), prove that the curvature is given by dr p2 + 2 do () dr - r de? K = 3 2 dr () p2…

A:

Q: 1) Prove Green's Theorem for F = 3xy'i - 4xj around the curve C bounded %3D 160 by y = 2x and y =…

A: These questions are about green's theorem around any given curve c.

Q: 5) Let F(x, y, z) = (z − x², xz, y²) and C be the curve of intersection between x² +3y² 1 and z = x²…

A: Let us first parameterize curve C.

Q: Laplace Hansfolm of funchion: FCt)= Cast.u(t-7), Whele u(t-I) is a unit funchon.

A:

Q: ® Use Craumoy's rule to solre fhe following linear syslems: rele to solve al fxy=0 8? S x+2y-2 = 0…

A:

Q: 8. Given the space curve C as the image of the vector valued function r(t) = 2ti+t³j+-t-¹ k OR r(t)…

A:

Q: Compute the mean curvature for u = 1 and v= 0.

A: Given,           X(u, v)=u2-cosv, u+sin t, cosv To find the mean curvature for u=1, and v=0.

Q: Evaluate ∮C (x + 3y)dx + ydy where C is the Jordan curve given by the graphs of y = e^x, y = e^−x…

A: ∮Cx+3ydx+ydy Where C:y=ex, y=e-x, y=e-1

Q: If F(x, y, z) = z - f(x, y) and s is the graph of z = f(x, y) and (a, b,c) is a point on the surface…

A:

Q: solve for the following using the concepts of vector calculus:

A:

Q: tind Yaduis ot curuature of the Curue O r= 2t i t (1+); - 2t k at (2,1,-2) 13 Brofi+tietk at the…

A:

Q: Consider the curve C parametrized by x = 19 – t and y = 17 for all real t. %3D Then C is a O…

A:

Q: d (x, y) = (x - y)^2 does the function define a metric o

A:

Q: Show X(t) where d that for Curve lacost, asint, at Cot a) constant, is the ratio k is Cant Constant.

A: