Vectors Homework Help

a-5) Write the equation used to find the locations of the extrema of f(x,y)= xy + 2xy³ on a circular disk given by x² + y² ≤9 using the method of substitution. Do not solve for the extrema. b-) A cylindrical metal can is to be manufactured from a fixed amount of sheet metal. Use the method of Lagrange multipliers to determine the ratio between the dimensions of the can with the largest capacity.

2. An airplane wishes to travel from Calcland to Vectorville. The true bearing of Vectorville from Calcland is 330°. The wind has a velocity of 100 km/h[W40°S]. The plane has an airspeed of 450 km/h. The pilot sets the heading so that the plane will travel to Vectorville. a) Determine the groundspeed of the plane. b) Determine the heading that the pilot must take.

Problem 1- If size of the position vector is R, a) calculate: V(R) b) calculate: V(R") \text { Note: Position vector is } \mathbf{R}=x_{i} \hat{e}_{i} \text { and } R^{2}=x_{i} x_{i}

\text { 1: Evaluate the line integral } \int_{F} \cdot d \vec{r} \text {, where } C \text { is given by the vector function } \vec{r}(t)

2. A vector field is given by A(x, y, z)=A_{0} \hat{\mathbf{z}}+A_{1} \frac{\mathbf{r} \times \hat{\mathbf{z}}}{r^{2}} where A0 and A1 are constants, and r=\sqrt{x^{2}+y^{2}+z^{2}} (a) Calculate div A. (b) Calculate curl A.

Consider the box shown below, where 0 <a < 2,0<y<1, and 0 <=< 1. \text { The rectangular prism is subjected to an electric field given by } \mathbf{E}=-7 x^{2} \hat{\mathbf{i}}+7 x^{2} y^{2} \hat{\mathbf{j}}+6 x z^{2} \hat{\mathbf{k}} Find the flow through the side where r = 2. \underline{\mathbf{E}} \cdot \underline{\mathbf{n}} d S= Enter these as, e.g. *dx, *dy, and/or *dz. \iint_{\text {rightside }} \mathbf{E} \bullet \underline{n} d S= Find the flow through the top side. \underline{\mathbf{E}} \bullet \underline{\mathbf{n}} d S= \iint_{t o p} \underline{\mathbf{E}} \bullet \underline{\mathbf{n}} d S= Use Gauss's Theorem to calculate the total flow out of the box. \operatorname{div}(\underline{\mathbf{E}})= \oint \oint_{b o x} \mathbf{E} \bullet \underline{\mathrm{n}} d S=

5. Find parametric equations of the line of intersection between the planes 2 x+3 y+4 z=5 \text { and }-x+y-z=1

1. Verify the divergence theorem for the field A= 2r3z and the volume region given by 0 < x < 1, 1< y < 2, 2 < z < 3.

Calculate the 5th roots of -3+j3 in polar and exponential forms.

3. [4 points] Let X be a vector field on Rm such that for every p = Rm the maximal integral curve of X going through p at time zero is defined on the whole of R. Consider the stereographic projection : Sm - {N} → Rm, where N denotes the North Pole. Let Y be a vector field on the sphere Sm such that its restriction to Sm{N} equals (0-¹),X. What can you say about YN, the value of Y at the North Pole?