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- Let S be the surface defined by the vector function R(u, v) = (u cos v, u – v, u sin v) with u E R and v E [0, 27]. - a. Find the equation of the tangent plane to S where (u, v) = (2, 7). b. Determine the area of the portion of S where 0 < u<1 and 0 < v< 4u.arrow_forwardConsider the following function. T: R2 → R, T(x, y) = (2x2, 3xy, y?) Find the following images for vectors u = (u,, u2) and v = (v,, v2) in R2 and the scalar c. (Give all answers in terms of 1' "1, U2, V1, V2, and c.) T(u) T(v) T(u) + T(v) = T(u + v) CT(u) = T(cu) = Determine whether the function is a linear transformation. O linear transformation not a linear transformationarrow_forwardFind the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter (Use technology to sketch) x2 + y? + z? = 10, x + y = 4 x = 2 + sin (t)arrow_forward
- 2. The vector function R(u, v) = (sin u cos v, sin u sin v, cos u), (u, v) E [0, 1] × [0, 27] gives a unit sphere. Use double integrals and the above parametrization to find surface area of a unit sphere.arrow_forwardPlease recheck and provide clear and complete step-by-step solution in scanned handwriting or computerized output thank youarrow_forwardLet f : R2 → R differentiable with f(a, b) = 5, ∂f/∂x (a, b) ≠ 0 and ∂f/∂y (a, b) ≠ 0. Let C be the intersection curve of the equation f(x, y) = 5 and u→ a nonzero vector tangent to C at the point (a, b). Detail whether each of the following statements is true or false: I. ∂f/du→ ≠ 0 necessarily. II. ∇f(a, b)·u→ = 0, necessarilyIII. If v→ is any vector and w→ = ∇f(a, b), then we necessarily have ∂f/dw→(a,b)≥ df/dv→(a,b). IV. r(t) = (a + df/dx (a,b) t, b - df/dy (a,b) t), t ∈ R, is the equation of the line normal to C at (a,b). V. If γ(t) = (x(t), y(t)), t ∈ I, where I is some open interval, is a parametrization of C and h(t) = f(γ(t)), then h'(α) = 0, where α ∈ I is such that γ(α) = (a, b).arrow_forward
- Let S be the surface defined by the vector function R(u, v) = (u cos v, u – v, u sin v) with u ER and v E (0, 27]. %3D a. Find the equation of the tangent plane to S where (u, v) = (2, E). b. Determine the area of the portion of S where 0arrow_forwardLet the temperature T in degrees at the point (x,y,z), with distances measured in cm, be T(x,y,z)=3x – 4y+3z. Let q be the real numbe such that the rate at which the change in temperature at (-2,0,3) per unit change in the distance travelled in the direction of the vector (1,q,1> is 4°/cm. Find q. (Note that the "direction" of a vector is always a unit vector "pointing the same way.") none of the other answers -7/40 17/56 1/12 1/2arrow_forwardLet F= w x R, where w (omega) is a constant vector. Calculate the integral of the dot product F and dRarrow_forwardAssume that u and u are continuously differentiable functions. Using Green's theorem, prove that Uz JE v|dA= [udv, Uy D where D is some domain enclosed by a simple closed curve C with positive orientation.arrow_forwardShow that the vector-valued function shown below describes the function of a particle moving in a circle of radius 1 centered at a point (5,5,3) and lying in the plane 3x+3y-6z = 12arrow_forward42. Derivatives of triple scalar products a. Show that if u, v, and w are differentiable vector functions of t, then du v X w + u• dt dv X w + u•v X dt dw (u•v X w) dt dt b. Show that d'r dr? dr dr d'r dt dt r. dt dr? (Hint: Differentiate on the left and look for vectors whose products are zero.)arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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