Mathematics

Step 1: Define an even function and the Fourier coefficients. An even function satisfies . The Fourier series is . - - -
Step 2: Analyze the coefficients for an even function. - and : The cosine function, , is also an even function. The product of two even functions ( and ) is even. The integral of a non-zero even function over a symmetric interval (like to ) is generally non-zero. Thus, for a general even function, we expect and . Statements A and B are generally correct. Statement C is incorrect. - : The sine function, , is an odd function. The product of an even function ( ) and an odd function ( ) is an odd function. The integral of any odd function over a symmetric interval is always zero. Thus, for any even function, . Statement D is correct.
Step 3: Conclude the correct statements. The correct statements describing the Fourier coefficients for an even function are that and can be non-zero, while must be zero. This corresponds to statements A, B, and D.

Step 1: Establish the relationships between Cartesian and polar coordinates. We are given and . The inverse relationships are and .
Step 2: Calculate the partial derivatives. - A. : Differentiate with respect to : . This matches III. - B. : Differentiate with respect to : . This matches II.
Step 3: Calculate the Jacobians. - C. :} This is the determinant of the Jacobian matrix for the transformation from polar to Cartesian coordinates. This matches IV. - D. :} This is the inverse of the Jacobian calculated in C. This matches I.
Step 4: Formulate the correct matching sequence. The matches are: A III, B II, C IV, D I. This corresponds to option (2).