Writing a Tangent PlaneIn Exercises 57 and 58, show that the tangent plane to the quadric at the point
Hyperboloid:
Tangent Plane:
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Calculus
- Flux across hemispheres and paraboloids Let S be the hemispherex2 + y2 + z2 = a2, for z ≥ 0, and let T be the paraboloid z = a - (x2 + y2)/a, for z ≥ 0, where a > 0. Assume the surfaces have outward normal vectors.a. Verify that S and T have the same base (x2 + y2 ≤ a2) and thesame high point (0, 0, a).b. Which surface has the greater area?c. Show that the flux of the radial field F = ⟨x, y, z⟩ across S is 2πa3.d. Show that the flux of the radial field F = ⟨x, y, z⟩ across T is 3πa3/2.arrow_forwardUsing Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.F = (x - y) i + (x + y) j; C is the triangle with vertices at (0, 0), (7, 0), and (0, 6) a) 0 b) 252 c) 84 d) 42arrow_forwardWrite the parametric equation of the bilinear surface corresponding to four points P(0 0)= 0.25 0 P(1 0)- 0.75 0 P(0.1)- 0.75 0.9 PEL.A F 0.25.0.8 .Also calculate the tangent and normal vectors at the mid-point of the surface.arrow_forward
- Classify, according to type (hyperbolic, elliptic, parabolic) the equations(i) Uxx +2Uxy +Uyy −Ux +U =0. (ii) 2Uxy +Uy +Ux =0.(iii) Uxx − Uxy − 2Uyy = 0.arrow_forwardNeat handwriting is perfectly acceptable. Please submit your assignment as ONE PDF file. Find the scalar equation of the plane containing both the line of intersection of the planes defined by ?1: 2x + 3y + z – 2 = 0 and ?2: x + 2y – z – 5 = 0 and the point P (1, 0, –2)arrow_forwardFinding Curvature In Exercises 23-28, find the curvature of the plane curve at the given value of the parameter. r(t)=(t, sent) , t=pi/2arrow_forward
- Calculus In Exercises 43-46, let f and g be functions in the vector space C[a,b] with inner product f,g=abf(x)g(x)dx. Let f(x)=x+2 and g(x)=15x8 be vectors in C[0,1]. aFind f,g. bFind 4f,g. cFind f. dOrthonormalize the set B={f,g}.arrow_forwardFill in the blanks. A is the intersection of a plane and a double-napped cone.arrow_forwardFinding Inner Product, Length, and Distance In Exercises 35-38, find a p,q, b p, c q, and d d(p,q)for the polynomials in P2using the inner product p,q=a0b0+a1b1+a2b2. p(x)=1+x2, q(x)=1x2arrow_forward
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