PART A ] (1) Find a complex potential function g(z) of the given vector field F (x,y). (2) find the equation of a streamline of the given vector field F (x,y). F(x,y) = < 4x² + 5x – 4y² +9, – 8xy – 5y +1 >
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- F = (x² - y)i + + (4z)j + (x²)k Find the curl (curl calculation) of the vector field?1) Find the domain of the vector-valued function r(t) = (3t+1,6,ln(9-t)).A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v=(x-y,z+y+7,z2) and the net is decribed by the equation y=1-x2-z2, y20, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.)
- A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v = (x-y, z + y + 3, z²) and the net is decribed by the equation y = √1-x²-2², y ≥ 0, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.)Match the vector field with its graph. F(x, y) = yi -6- 事 5 -5+3. Let f(x, y) = sin x + sin y. (NOTE: You may use software for any part of this problem.) (a) Plot a contour map of f. (b) Find the gradient Vf. (c) Plot the gradient vector field Vf. (d) Explain how the contour map and the gradient vector field are related. (e) Plot the flow lines of Vf. (f) Explain how the flow lines and the vector field are related. (g) Explain how the flow lines of Vf and the contour map are related.
- Determine if each of the following vector fields is the gradient of a function f(x, y). If so, find all of the functions with this gradient. (a) (3x² + e¹0) i + (10x e¹0 - 9 siny) j (b) (10x el0y 9 sin y) i + (3x² + e¹0y) j a) I have placed my work and my answer on my answer sheetA net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by = (x - y, z + y + 9, z) and the net is decribed by the equation y = V1-x - z, y 2 0, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.) V. dS =(a) Find the direction for which the directional derivative of the function f(x, y, z)=√x² + y² + z² is a maximum at P = (8,5, 2). (Use symbolic notation and fractions where needed. Give your answer in vector form.)
- (b) Let r(t)ti-j+(4-e) k, then the vector equation of the line tangent to the graph of r(t) at the point (4, 1,0) on the curve is A.r-(4i+ j) + t(-4i+j+4k) B. r-(4i- 3) + t(-41+j+ 4k) (e) TheGraph the vector field F(x,y)=(x-y)i+xjThe figure shows the vector field F (x,y) = {2xy,x2 } and three curves that start at (1,2) and end at (3,2). (a) Explain why∫c F. dr has the same value for all the three curves. (b) What is this common value?