2. Let u € C² (D), where D is the unit disk with center at origin in R2. Suppose for any smooth function w with w = 0 on OD, we have Į D wAu dV = 0. Is u harmonic in D? If yes, prove it. If no, write down an example.

Q: O A. Choose the correct sketch below. A) 16 225- If 15 30 Q Q A5 225 O B. Ay Ay 30- 15- g t 225 Q Q…

A: To plot the parametric curve in the XY plane

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Q: Q4. What is the Laplacian of the following vector function? v = x²x+ xz°ŷ +y° î

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A: Solution:

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A: The natural point that comes in our mind is (2,6).

Q: Find T, N, and k for the space curve r(t) = (5 sin t)i + (5 cos t) j+ 12tk. T(t) =i+j+k

A: We have to find  and .

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A:

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Q: ASSIGNMENT NO.2 = 1. If F = 3xyi - y²j, evaluate fF.dr, C is the curve in the xy-plane, y to (1,2)…

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Q: Let F(x, y) = (x¹7e², 7) and C be the path along the right half of the circle (r - 3)² + (y – 5)² =…

A: 28.2) To evaluate the line integral ∫CF.dr for the function F(x,y)=(x17ex2,7) where C is the right…

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A:

Q: and C be th = = 0. If [F.d.

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Q: Question 4. Let o(u, v): R² →→→ R³ be a smooth function (not necessarily a surface patch). Let E =…

A: We have to prove that σu×σv2=detDσTDσ=EG-F2 where,  E=σu.σu, F=σu.σv, and G=σv.σv

Q: Let C be the parametric curve defined by æ(t) = t? – 2, | y(t) = t3 – 3t + 1. Determine if C is…

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Q: 1. Let R be a vector function such that T(t): = =(cos 2t,sin 2t) and R(0) = (1,0.-1). Find i. The…

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Q: Let (t) = (t, t, tº, tª) for all t € R. Define C = Þ(R). Let p = (1, 1, 1, 1). Derive the tangent…

A: Let  for all Let the point be We need to calculate the tangent and normal spaces to C at p.

Q: Find T, N, and k for the space curve r(t) = i+Tj, t> 0. T(t) = (O i+ (Oi

A:

Q: tion z2 – (x2 + y2)2 and let p = (0, 0, 0). ermine the set Tp(Zf ) of all tangent prs to Zf at p. Is…

A: Sol

Q: Let f(x, y) = x²y – xy. Evaluate | Vf•dr where C is any oriented curve from (4, –1) to (1,5).

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Q: 3. Consider the circle as a vector-valued function r(t) = cos(t)i + sin(t)j. Calculate limt→+∞ r(t)…

A: Find the limit

Q: 2. Let C be the graph of R(t) = (t sint, t³). a. Using the definition, show that Ŕ(t) is continuous…

A:

Q: Let C be the parametric curve defined by x(t) = t² – 2, y(t) = t³ – 3t + 1. Determine if C is…

A: Given that,A parametric curve C is defined byx(t)=t2-2y(t)=t3-3t+1The concavity of the parametric…

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Q: sketch the space curve represented by the vector- valued function and give the orientation of the…

A: The given curve are (i) r(t)=-t+1i+4t+2j+2t+3k. (ii) r(t)=t,t2,23t3. To Sketch: The given curves…

Q: 2. Let C be the graph of R(t) = (t sint, t³). a. Using the definition, show that Ŕ(t) is continuous…

A: