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2. (a) (4 marks] State whether the divergence of the vector field shown below is on...
a) The following vector field State whether the divergence of at point A is positive, negative or zero. b) Say if the rotational of at point B is a null vector, which points in the direction of the z-axis or points in the negative direction of z. We were unable to transcribe this image履 2 0 2 4 We were unable to transcribe this imageWe were unable to transcribe this image 履 2 0 2 4
(a) Prove that the divergence of a curl is always zero for any vector field. (b) Prove that the curl of the gradient of a scalar function is always zero.
Find the divergence and curl of the vector field \(\vec{F}=s^{\frac{1}{2}} \hat{\phi}\)s20
Exercise. Below we have plotted a discrete sampling" of a vector field -4 -2 2 Let C bea circle of radis3centered at the origin n drawn in a counterclockwise fashion. What conelasions seem to he true? This is a gradient field. This is not a uruient Gelkl. This field has positive curl. This field has negative e curl. oF.dp- k.P.dp > ถ ? Check work Exercise. Below we have plotted a discrete sampling" of a vector field -4 -2 2...
Exercise. Below we have plotted a discrete "sampling of a vector field: -2 2 4 Let C be a circle of radins 3 centered at the origin drawn in a counterclockwise fashion. What concusions seem to be true? This is a gradieut field This is not a gradient field. This field has positive cur This field bas negative curl. c F.dp X Try again Note that the raclias of the circle is irreverent. Exercise. Below we have plotted a discrete...
IL. Displacement field due to the divergence of the polarization nofreechargeright has a fixed, radial polarization? = CPas shown, but has no free charge A. Consider the sources of the displacement field. 1. Does D have non-zero divergence at any point? If so, where? 2. Does D have non-zero curl at any point? If so, where? 3. Given your answers above, what direction does D point outside the sphere? If D is zero outside the sphere, state so explicitly. Explain...
Exercise. Below we have plotted a discrete "sampling" of a vector field: 40 -2 2 Let C be a circle of radius 3 centered at the origin drawn in a counterclockwise fashion. What conclusions seem to be true? This is a gradieut field This is not a gradient field This field has positive curl. This field has negative curl. cF.dp>o ? Check work Exercise. Below we have plotted a discrete "sampling" of a vector field: 40 -2 2 Let C...
9. A vector field is defined as: 2 marks (a) Sketch this field on the below axis. 0.5 0.5 0.5 2 marks (b) Evaluate ▽ . F. (c) Evaluate ▽ × F and hence conclude whether F is a conservative vector field 2 marks (d) Demonstrate whether there is a scalar function φ such that F and 3 marks] hence conclude whether Fis a conservative vector fiel Recall dx and that this can be used to determine tan dx 9....
Consider the following region R and the vector field F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in the flux form of Green's Theorem and check for consistency. c. State whether the vector field is source free. F = (8xy,9x2 - 4y?); R is the region bounded by y = x(6 - x) and y=0. .- a. The two-dimensional divergence is 0 b. Set up the integral over the region. dy dx 0 Set...
2. (12 points) Determine whether the following statements are true or false. Explain why, or provide a counterexample (a) For conservative vector fields, the divergence is always zero. (b) The circulation of a vector field along a closed curve is different depending on the orientation of the curve. (c) If the curl of a vector field at the origin is 2,0,1), then the average circulation around the y axis at the origin is counterclockwise.