Mathematical working with explanatory statements

ENM2600 Assignment 2, value: 15%, Submit your assignment via EASE as a single file in doc- or pdf-format before the deadline. Please check your assignment carefully before submitting. Hand-written work is acceptable, provided it is presented in the neat and legible form. When you complete your assignment, submit a scanned copy via EASE and keep the original in a safe place until your assignment is marked and returned to you. You may also type-set your answers, if your software offers quality notation. In any case please use correct notations for mathematical symbols, otherwise you may lose marks. This assignment will be assessed not only for its mathematical content, but also for clarity of communication. Link your mathematical working with explanatory statements, use good English and write in grammatically correct sentences. Look at the worked examples in the Text Books and Study Book, as well as the sample solutions in the Lecture slides to see the level you should 1 aim at. Up to 15% of the marks may be deducted for poor language and notation. If a student submit assignment after the due date without prior approval of the Examiner, then a penalty of 5% of the total marks gained by the student for the assignment may apply for each calendar day late up to ten working days at which time a mark of zero may be recorded. No assignments will be accepted after model answers have been posted on Study Desk. Question 1. [10 marks] The surface is defined by the following equation: z(x, y) = (3 + cos 2x)e 1+y 3 + 5xy2 − 6. Find the equation of the tangent plane to the surface at the point P(0, −1). Question 2. [15 marks] Find all critical points of the function f(x, y) = ln (x + y) + x 2 − y + 64 and determine their character, that is whether there is a local maximum, local minimum, saddle point or none of these at each critical point. 2 Question 3. [20 marks] Calculate the double integral by transferring to polar coordinates: I = ∫ 3 0 dx √ 9−x2 ∫ 0 ln (1 + x 2 + y 2 ) dy. Make a sketch of the domain of integration. Present your answer in exact form and then evaluate it using a calculator up to two decimal places. Question 4. [20 marks] Evaluate the work done by the vector field I = ∫ Γ [( y x 2 + y 2 − 1 ) dx − x x 2 + y 2 dy] between the points A(π/6, π/6) and B(−π, π). Make an independent decision about the most convenient path of integration Γ. Present your answer in exact form and then evaluate it using a calculator up to two decimal places. Hint: First find the vector field and determine whether it is conservative or not. Then make a decision of how this integral can be evaluated. Question 5. [20 marks] Find the total electric charge Q of the plate D : {0 ≤ x ≤ π; 0 ≤ y ≤ π/2} if the surface density of the charge is: q(x, y) = x sin (x + y). 3 Present your answer in exact form and then evaluate it using a calculator up to two decimal places. All quantities are presented here in the dimensionless form. Question 6. [25 marks] Find the total mass M of the wire if its line density is ρ(x, y, z) = x 2−y 2+z 2 . The vector equation of the wire is r = 4 costˆi + 4 sin tˆj + (3t − 5) kˆ, where 0 ≤ t ≤ π/4. Present your answer in exact form and then evaluate it using a calculator up to two decimal places. All quantities are presented here in the dimensionless form. Question 7. [25 marks] Evaluate the integral of the vector field F = 5 x yz ˆi + 3 y xz ˆj + 7 z xy kˆ along the line Γ: r = 2 costˆi − 3 sin tˆj + cost kˆ, where t varies from π/6 to π/4. Present your result in exact form and evaluate it up to two decimal places. 4 Question 8. [15 marks] For the given vector field F = (x 2 − y 2 )ˆi + 4xyˆj + (x 2 − xy) kˆ compute: (a) the divergence in the point P(1, 2, 3); [5 marks] (b) curl in the point P(1, 2, 3). [10 marks] [Total: 150 marks] ——————– End of Assignment 2 ——————– 5