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- Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?Use sine and cosine to parametrize the intersection of the cylinders x² + y = 81 and x? + z? = 81. Find a vector parametrization r(t) = (x(1), y(t), z(1)). (Use symbolic notation and fractions where needed. Use cosine for parametrization x variable.) x(t) = y(t) = + z(t) =Find the vector equation that represents the curve of intersection of the cylinder z + y = 16 and the surface z = ze". Write the equation so the r(t) term contains a cos(t) term. z(t) %3D y(t) z(t) =
- Find the scalar component of H tangential to the plane z=0The vector equation r (u, v) = u cos vi + u sin vj + vk, 0 < v < 5x, 0 < u < 1, describes a helicoid (spiral ramp). What is the surface area? 10.25pi1. Let R and b be positive constants. The vector function r(t) = (R cost, R sint, bt) traces out a helix that goes up and down the z-axis. a) Find the arclength function s(t) that gives the length of the helix from t = 0 to any other t. b) Reparametrize the helix so that it has a derivative whose magnitude is always equal to 1. c) Set R = b = 1. Compute T, Ñ, and B for the helix at the point (√2/2,√2/2, π/4).
- 2. Calculate the Jacobian of the coordinate change (x, y, z) to cylindrical coordinates (r, 0, z).Q. For Ē=3x⸠-xyâ, , calculate (Ē- dĩ y from A(-2,0) to B(2,0), C2 (0,2) a) along the path of C1 b) along an are of a C2, by converting Ẽ to cylindrical coordinate system. c) Is line integration independent of the shape of the path, explain. A (-2,0) C, B (2,0)(b) Solve the electric flux through a paraboloid z = x² + y² – 16 that lies below plane - z = 0. It is oriented by an outward unit normal vector with F = 3xî + 4yĵ – 5z²k.
- Consider the following function. T: R? - R?, T(x, y) = (3x², 3xy, y²) Find the following images for vectors u = (u,, u2) and v = (v,, v2) in R? and the scalar c. (Give all answers in terms of u,, uz, Va. and c.) %3D T(u) = T(v) = 3u + 31. T(u) + T(v) = T(u + v) - CT(u) = T(cu) = Determine whether the function is a linear transformation. linear transformation not a linear transformationLet u = 2i - j, v = 3i + j, and w = i + 4j.Find specified scalar (4u) . v.