Exercise 2.6.7. Let f: M→ N be a smooth map between manifolds. Prove that the tangent map TM →TN: (p, v) → (f(p), df (p)v) is smooth.

Q: a. Show that n(t) = -g'(t)i + f(t)j and - n(t)=g'(t)i-f'(t)j are both normal to the curve r(t) =…

A: Using n(t) = - g'(t)i + f(t)j and - n(t) = g'(t)i – f'(t)j are both normal to the curve r(t) =…

Q: Let A be an m x n matrix, and let B and C have sizes for which the indicated sums and products are…

A: If the product AB is defined, the entry in row i and column j of AB is the sum of the products of…

Q: 1) Show that F is conservative the find a function f such that F = Vf and use that function to…

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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 .

Q: Prove the dot product rule. If C₁ (t) and C₂ (t) are differentiable curves in R" then: d [c₁ (t) ·…

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Q: The two curves in the three dimensional space IR3 intersect at the point P 1) The tangent vector at…

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Q: Use software to plot the function f(x, y) = x² + y² with the intersecting planes x = 0 and z = 4.…

A: Perspective 1: 3D Plot of  f(x,y)=x2+y2In this view, we see the 3D surface plot of the function…

Q: 3. Consider the curve C defined by the vector-valued function R(t) = -2):+-3)j+ cos(t – 2)k, - oo…

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

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Q: let T: P4(R)→R2 be the map given by T(f)=(f(1),f"'(1)) where P4(R) is the set of polynomials of…

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Q: Consider the curve C = C₁ U C2 U C3 U C4, piecewise smooth, oriented from the point A (3, 7, 3) to…

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Q: Find T, N, and K for the space curve r(t) = 7 T(t) = N(t) = t³ +1 İ+ İ+ 1 +1 j +4 4j, t>0.

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Q: Consider the curve G given by the vector function f(t) = tỉ + tj + (2 +t²)k where t e [0,4]. Give an…

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

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Q: Consider F and C below. F(x, y, z) = yzi + xz j + (xy + 14z) k C is the line segment from (2, 0, -2)…

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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. Let C be the curve with parametrization given by r(t) = (t − 1)i +3t²j + tk for t € R. The…

A: Given  r(t)=(t-1)i +3t2j +tKand the point  (0,3,1) corresponds to t=1then drdt  can ber ,(t)…

Q: Consider F and C below. F(x, y, z) = (y²z + 2XZ² ) Î + (2xyz)j + (xy² + 2x² Z) K₂ C² X=√E, y = t +…

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Q: Consider F and C below. F(x, y, z) = yz i+xz j + (xy + 10z) k C is the line segment from (2, 0, -1)…

A: I have explained everything with in the solution. Please go through that.

Q: Consider F and C below. F(x, y, z)= yzi + xz j + (xy + 12z) k C is the line segment from (2, 0, -1)…

A: The given vector function: F(x, y, z)=yz i+xz j+(xy+12z) kGiven line segment:  C: from (2, 0, -1) to…

Q: Evaluate F ds (or F ds) where Jc F = (y – x2, z - y², x – z) and C is the curve r = joining (0,0,0)…

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Q: 1. Let F(x, y) = (1 — y, ½x³ +e³²), and let C with the position vector r be the portion of the…

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Q: Consider the piecewise smooth curve C = C1 u C2 u C3 u C4, oriented from the point A(1, 1, 2) to…

A: Given that, F=4x3-2z, 3y2,-2x Find the value of ∫cF. dr we have to find the function f such that…

Q: Consider F and C below. F(x, y, z) = yz i + xz j + (xy + 16z) k C is the line segment from (1, 0,…

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Q: the space curve (t) = (2t, t², 2t)

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Q: Exercise 1. Define G: R² → R by G(x, y) = 0, if (x, y) = (0,0); x²y x² + y² , if (x, y) = (0,0). 1)…

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Q: Let C2 be the curve defined by R(t) = (t² +t + 3, 4t – 5, 2+ t – t2), t e [0, 1]. %3D | (x + y+ 2)…

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Q: Let C: [o,2]> R be a curve defined by C(t) = (t²,t'). Evaluate. Sy dx x dy.

A: Given:  C: [0, 2]→ℝ2   is a curve defined by                C(t)=(t2, t3)implies x=t2 and…

Q: 13. Let F (x, y) = (1(²y²)x, af(x²y? , a(y²) Y) and suppose that C is a smooth curve parametrized…

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