This PDF document contains an assignment for "Linear Algebra and Calculus," assigned by Chaitanya Bharathi Institute of Technology for the course coded 22MTC01. There are 20 questions in total, covering various topics in calculus and linear algebra, such as partial derivatives, functional dependence, Taylor series expansions, optimization, surface angles, directional derivatives, vector fields, and integral theorems (like Green’s Theorem).

Would you like a solution to a specific question, or detailed explanations on any of the problems? Here are some related inquiries you might consider:

1. How do you solve for partial derivatives in complex functions, as in Question 1 or 2?
2. What approach would you use to find the maxima and minima of a multivariable function, as asked in Question 7?
3. How can you determine if functions are functionally dependent, as requested in Question 5?
4. What are the steps to evaluate an integral using Green's Theorem, as in Questions 18-20?
5. How do you calculate the directional derivative along a specified path, like in Question 10?

Tip: When solving problems involving directional derivatives, always start by determining the unit vector for the direction you're considering. This simplifies calculations and ensures accuracy in direction-based rates of change.