Find the divergence of the vector field. (V.V=V + OV₂ + 2/₂V₂) дх дz V(x, y, z) = (x²y, 3y, 5) = x²yî + 3yĵ + 5k
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- Compute the flux of the vector field F = 2zk through S, the upper hemisphere of radius 6 centered at the origin, oriented outward. flux =Question 1 Four stationary electric charges produce an electric field in space. The electric field depends on the magnitude of the test charge used to trace the field O has different magnitudes but same direction everywhere in space is constant everywhere in space has different magnitude and different directions everywhere in space CANADConsider a triangle in the presence of a uniform electric field given by 5.6 i N/C. The endpoints of the triangle are: (0 m, 0 m, O m), (7 m, O m, 4 m), and (2 m, 3 m, O m). Determine the absolute value of the electric flux through the triangle. Give your answer in units of N-m?
- Consider the vector field ʊ(r) = (x² + y²)êx + (x² + y²)êy + z²êz. Decompose the vector field (r) into the sum of two other vector fields, a (r) and 5(r), such that a(r) has no divergence (it is solenoidal) and 5 (r) has no curl (it is irrotational). The answer is not unique. This is the Helmholtz decomposition.Evaluate the line integral, where C is the given curve. Sc (x + yz)dx + 2x dy + xyz dz C consists of line segments from (1, 0, 1) to (2, 2, 1) and from (2, 2, 1) to (2, 4, 3). The force exerted by an electric charge at the origin on a charged particle at a point (x, y, z) with position vector r = is F(r) = Kr/1r|³ where K is a constant. Find the work done as the particle moves along a straight line from (5, 0, 0) to (5, 1, 5). Find the mass and center of mass of a wire in the shape of the helix x=t, y = 5cos(t), z = 5sin(t), 0 st ≤ 2π, if the density at any point is equal to the square of the distance from the origin. (mass) ) (center of mass)Compute the flux of the vector field F=2xi+2yj through the surface S, which is the part of the surface z=36−(x^2+y^2) above the disk of radius 6 centered at the origin, oriented upward. flux = _____
- Calculate the flux of the vector field F(x, y, z) = (5x + 8)i through a disk of radius 7 centered at the origin in the yz-plane, oriented in the negative x- direction.= Let er be the unit radial vector field. Compute the outward flux of the vector field F er/r² through the ellipsoid 4x² + 6y² + 9z² = 36. [Hint: Because F is not defined at zero, you cannot use the divergence theorem on the bounded region inside of S. ]Calculate the flux of vector field F = (xy°, x²y) across the circle of radius 1 centered at coordinates (0, –1).
- Suppose a unidirectional vector field E exists across an enclosed surface A as shown below. which among the following is/are true? 153 E The divergence volume integral value will only be equal to 0 if E is a uniform field. O The divergence volume integral value will be positive if the magnitude of E is increasing along the positive z-hat direction. The divergence volume integral value will always be equal to 0. Only the top and bottom cylindrical phases have non-zero divergence surface integral values.Calculate the flux of the given vector field by evaluating the line integral directly alongthe given curve for the below parts:(a) The vector field is ⃗ F = (x − y)⃗i + x⃗j. The curve is the circle x^2 + y^2 = 1in the xy-plane. Use the parameterization x = cos t and y = sin t.(b) The vector field is ⃗ F = (x − 1)⃗i + y⃗j. The curve is a circle of radius 3centered at (1, 1). The parametric form of this circle is⃗r = (1 + 3 cos t)⃗i + (1 + 3 sin t)⃗j, 0 ≤ t ≤ 2π(c) The vector field is ⃗F = x⃗i + y⃗j. The curve is the line segment from thepoint (0, 1) to the point (1, 3).Problem 3: UP 6.53 Charge is distributed uniformly with a density p throughout an infinitely long cylindrical volume of radius R. Show that the field of this charge distribution is directed radially with respect to the cylinder and that E(s) = ps 2€0 PR² 2€ S S≤R SZR