Evaluating a Line Integral of a
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Calculus
- Sketch the vector field F. O O -2 F(x, y) = i + (y − x)j + 1 X 2 Xarrow_forwardLet ø = p(x), u = u(x), and T = T(x) be differentiable scalar, vector, and tensor fields, where x is the position vector. Show that %3Darrow_forwardApplying the Fundamental Theorem of Line IntegralsSuppose the vector field F is continuous on ℝ2, F = ⟨ƒ, g⟩ = ∇φ, φ(1, 2) = 7, φ(3, 6) = 10, and φ(6, 4) = 20. Evaluate the following integrals for the given curve C, if possible.arrow_forward
- Subject differential geometry Let X(u,v)=(vcosu,vsinu,u) be the coordinate patch of a surface of M. A) find a normal and tangent vector field of M on patch X B) q=(1,0,1) is the point on this patch?why? C) find the tangent plane of the TpM at the point p=(0,0,0) of Marrow_forward人工知能を使用せず、 すべてを段階的にデジタル形式で解決してください。 ありがとう SOLVE STEP BY STEP IN DIGITAL FORMAT DON'T USE CHATGPT Find the integral of the vector function F(t)=(f.,cost)arrow_forwardWhat is the geometrical meaning of an integral of a vector function?arrow_forward
- Find r(t) u(t). r(t). u(t) r(t) = (4 cos(t), 9 sin(t), t – 4), u(t) = (18 sin(t), −8 cos(t), t²) Is the result a vector-valued function? Explain. Yes, the dot product is a vector-valued function. No, the dot product is a scalar-valued function.arrow_forwardSketch the vector field. x|y|F(x, y) = = 0 -4 2 02 -2 0 -2 2 2 -2 0 -3 -2 2 -2 -2 № → 3 2 1 -2 ین Y " X 2 1 2 3 +arrow_forwardDetermine the vector field of F. F(x, y) = yi - xj Oarrow_forward
- Consider the vector field ?(?,?,?)=(?+?)?+(2?+?)?+(2?+?)? F ( x , y , z ) = ( z + y ) i + ( 2 z + x ) j + ( 2 y + x ) k . a) Find a function ? f such that ?=∇? F = ∇ f and ?(0,0,0)=0 f ( 0 , 0 , 0 ) = 0 . ?(?,?,?)= f ( x , y , z ) = b) Suppose C is any curve from (0,0,0) ( 0 , 0 , 0 ) to (1,1,1). ( 1 , 1 , 1 ) . Use part a) to compute the line integral ∫??⋅?? ∫ C F ⋅ d r .arrow_forwardFind r(t) · u(t). r(t) = (5t – 3)i + t³j + 2k u(t) = t2i – 6j + t3k r(t) · u(t) = Is the result a vector-valued function? Explain. Yes, the dot product is a vector-valued function. No, the dot product is a scalar-valued function.arrow_forwardGravitational potential The potential function for the gravitational force field due to a mass M at the origin acting on a mass m is φ = GMm/ | r | , where r = ⟨x, y, z⟩ is the position vector of the mass m, and G is the gravitational constant.a. Compute the gravitational force field F = -∇φ .b. Show that the field is irrotational; that is, show that ∇ x F = 0.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage