Questions asked by Rocky

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• How to find the volume of the region bounded by the curves #z1=x^2+y^2 and z2=1−x^2−y^2# ?
  
• How to solve this question? Given that: # f(x,y,z)= x^3yz^2# Find # (partial ^2f)/(partial x^2)# and #(partial^2f)/(partial y^2)#. Show that #(partial^2f)/(partial x partial y)=(partial ^2f)/(partial y partial x)# ?
  
• Which appropriate trigonometric substitution is needed to evaluate: #intdx/sqrt(x^6-x^8)# ? Thank you so much in advance! Greetings
  
• How do you solve the equation 𝑤̅×<2,0,−2>=<4,0,4> where 𝑤̅=<𝑤1,𝑤2,𝑤3> is nonzero with a magnitude of 2?
  
• What is the dot product 𝑠̅.𝑝̅ (numerically) and what does this represent?
  
• Consider two general three-dimensional vectors 𝑢̅ and 𝑣̅. If the magnitude of 𝑢̅×𝑣̅ is 𝑘 times larger than the magnitude of 𝑢̅. Prove that the magnitude of 𝑣̅ is greater to or equal to 𝑘. How do you solve it?
  
• A line is inclined at equal angles to all three (x-, y- and z- axes) and passed through the origin. A second line passes through (1,2,4) and (0,0,1). What is the angle between the two lines?
  
• How to solve the equation 𝑤̅×<2,0,−2>=<4,0,4> where 𝑤̅=<𝑤1,𝑤2,𝑤3> is nonzero with a magnitude of 2?
  
• How to show the following are true for the two vectors 𝑢̅=4𝑥𝑦𝑖̅−2𝑧𝑗̅+(11𝑥^3)𝑘̅ ⁡and 𝑣̅=(𝑥𝑦/𝑧)𝑖̅+(3𝑥^2)𝑗̅ +6𝑘̅ ? a. ∇×(𝑢̅+𝑣̅)=∇×𝑢̅+∇×𝑣̅ b. ∇.(𝑢̅+𝑣̅)=∇.𝑢̅+∇.𝑣