Given the scalar function f(x,y,z) = x2 + y.

Welcome back — missed you this week. Given the scalar function f(x,y,z) = x^2 + y. Step 1: Find the gradient of f. The gradient of a scalar function f is given by: f = ( f)/( x) i + ( f)/( y) j + ( f)/( z) k First, calculate the partial derivatives: ( f)/( x) = ()/( x)(x^2 + y) = 2x ( f)/( y) = ()/( y)(x^2 + y) = 1 ( f)/( z) = ()/( z)(x^2 + y) = 0 Substitute these into the gradient formula: f = 2x i + 1 j + 0 k f = 2x i + j The gradient of f is 2x i + j. Step 2: Find the divergence. Divergence is defined for a vector field. Assuming the question asks for the divergence of the gradient of f, which is · ( f). Let F = f = 2x i + j + 0 k. The divergence of a vector field F = P i + Q j + R k is given by: · F = ( P)/( x) + ( Q)/( y) + ( R)/( z) Here, P = 2x, Q = 1, and R = 0. Calculate the partial derivatives: ( P)/( x) = ()/( x)(2x) = 2 ( Q)/( y) = ()/( y)(1) = 0 ( R)/( z) = ()/( z)(0) = 0 Substitute these into the divergence formula: · ( f) = 2 + 0 + 0 = 2 The divergence of the gradient of f is 2. Step 3: Find the curl. Curl is defined for a vector field. Assuming the question asks for the curl of the gradient of f, which is × ( f). Let F = f = 2x i + j + 0 k. The curl of a vector field F = P i + Q j + R k is given by: × F = (( R)/( y) - ( Q)/( z)) i + (( P)/( z) - ( R)/( x)) j + (( Q)/( x) - ( P)/( y)) k Here, P = 2x, Q = 1, and R = 0. Calculate the partial derivatives: ( R)/( y) = ()/( y)(0) = 0 ( Q)/( z) = ()/( z)(1) = 0 ( P)/( z) = ()/( z)(2x) = 0 ( R)/( x) = ()/( x)(0) = 0 ( Q)/( x) = ()/( x)(1) = 0 ( P)/( y) = ()/( y)(2x) = 0 Substitute these into the curl formula: × ( f) = (0 - 0) i + (0 - 0) j + (0 - 0) k = 0 i + 0 j + 0 k × ( f) = 0 The curl of the gradient of f is 0. Send me the next one 📸