Assignment 1.pdf

B.Tech. (CSE-AIML) Assignment Differential Calculus BMAS 0110 1. You are working as a design engineer in a materials science lab, where you are analyzing a new comp...

B.Tech. (CSE-AIML) Assignment Differential Calculus BMAS 0110 1. You are working as a design engineer in a materials science lab, where you are analyzing a new composite material's properties. The temperature distribution 𝜙 within the material is modeled by the equation 𝝓 = 𝟑𝒙𝟐 𝒚 − 𝒚𝟑 𝒛𝟐 This equation describes how the temperature varies based on the 𝑥, 𝑦, and 𝑧 coordinates within the material. Find the gradient ∇𝜙 of the temperature distribution 𝜙 at the point (𝟏, −𝟐, −𝟏). 2. Find the divergence and curl of the vector ̂ ⃗𝑽 = (𝒙𝒚𝒛)𝒊̂ + (𝟑𝒙𝟐 𝒚)𝒋̂ + (𝒙𝒛𝟐 − 𝒚𝟐 𝒛)𝒌 at the point (𝟐, −𝟏, 𝟏). 𝒙−𝒚 𝒙+𝒚 3. If 𝒖 = 𝒙+𝒚, and 𝒗 = show that they are not independent. 𝒙 Find the relation between 𝑢 and 𝑣. 4. As a computer science student working on a software project for data analysis, you are tasked with implementing an algorithm that estimates the outcome of small changes in input parameters. You decide to model the relationship between two parameters using the function 𝒇(𝒙, 𝒚) = 𝒚𝒙 where 𝑥 represents the rate of growth, and 𝑦represents a scaling factor. To estimate the function's value when 𝒙 = 𝟏. 𝟎𝟑 and 𝐲 = 𝟏. 𝟎𝟐. you plan to use a Taylor series expansion upto second degree terms around the point (𝟏, 𝟏). 𝝏𝒛 𝝏𝒛 5. If𝒇(𝒄𝒙 − 𝒂𝒛, 𝒄𝒚 − 𝒃𝒛) = 𝟎, show that 𝒂 +𝒃 = 𝒄. 𝝏𝒙 𝝏𝒚 6. If 𝒖 = 𝒙𝟑 + 𝒚𝟑 + 𝒛𝟑 + 𝟑𝒙𝒚𝒛, show that 𝝏𝒖 𝝏𝒖 𝝏𝒖 𝒙 +𝒚 +𝒛 = 𝟑𝒖. 𝝏𝒙 𝝏𝒚 𝝏𝒛 𝒙𝟑 +𝒚𝟑 7. If 𝒖 = 𝐭𝐚𝐧−𝟏 then verify 𝒖 is homogenous function and also 𝒙−𝒚 evaluate 𝝏𝒖 𝝏𝒖 a) 𝒙 +𝒚. 𝝏𝒙 𝝏𝒚 𝟐 𝝏𝟐 𝒖 𝝏𝟐 𝒖 𝟐𝝏 𝒖 𝟐 𝟐 b) 𝒙 𝟐 +𝒚 𝟐 +𝒛. 𝝏𝒙 𝝏𝒚 𝝏𝒛𝟐