Evaluating a Line
In Exercises 29-34, evaluate
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CALCULUS EARLY TRANSCENDENTAL FUNCTIONS
- 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_forwardSubject 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_forwardSketch the vector field F. O O -2 F(x, y) = i + (y − x)j + 1 X 2 Xarrow_forward
- Flux across curves in a vector field Consider the vector fieldF = ⟨y, x⟩ shown in the figure.a. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for 0 ≤ t ≤ π/2.b. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for π/2 ≤ t ≤ π.c. Explain why the flux across the quarter-circle in the third quadrant equals the flux computed in part (a). d. Explain why the flux across the quarter-circle in the fourth quadrant equals the flux computed in part (b).e. What is the outward flux across the full circle?arrow_forwardConsider 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_forward
- Question: Prove that the 2d-curl of a conservative vector field is zero, ( ∇ × ∇ f ) ⋅ k = 0 (here k is unit vector) for any general scalar function f ( x , y ).arrow_forwardConsider the vector-valued function r(t) = cos ti + sin tj + In(cos t)k. (a) Find the vectors T, N, and B of r at the point P = (1,0,0). (b) Find the tangential and normal components of the acceler- ation of r at the point P = (1,0,0).arrow_forwardConsidering the scalar functions ∅ = ∅ (x, y, z) and ψ = ψ (x, y, z), find the following expression? ∇. (∇ ∅ × ∇ ψ) =?arrow_forward
- Let 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_forwardSketch the plane curve represented by the vector-valued function and give the orientation of the curve. r(0) = cos(0)i + 6 sin(0)j O O -2 -2 y 5 -5 y -5 2 2 6 X X -6 -4 -2 -6 -4 -2 y 2 2 4 4 6 X Xarrow_forwardFind r(t) · u(t). r(t) = (7t – 4)i + j + 4k u(t) = t?i – 8j + t?k r(t) · u(t) = Is the result a vector-valued function? Explain. O Yes, the dot product is a vector-valued function. O No, the dot product is a scalar-valued function.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning