Calculus III Final Exam/ Final Exam - Multivariable Calculus With complete solution Newest RATED A+ 2025/2026 UPDATED 100% VERIFIED CORRECT

Calculus III Final Exam/ Final Exam - Mul variable
Calculus With complete solu on Newest RATED A+
2025/2026 UPDATED 100% VERIFIED CORRECT

Convert x to polar rcos(theta)



Convert y to polar rsin(theta)



convert r^2 to polar x^2+y^2



cos(2theta) iden es cos^2(theta) - sin^2(theta)

1-2sin^2(theta)

2cos^2(theta)



a dot b a1b1+a2b2



||a|| sqrt((a1)^2+(a2)^2)



a dot b (find angle between vectors) ||a||||b|| cos(theta)



what does orthogonal mean perpendicular



what does it mean is a dot b = 0 they are orthogonal



scalar projec on (b on a) (a dot b)/||a||

, vector project (b on a) ((a dot b)/||a||)(a/||a||)



Work (2 defini ons) ||F||||D||cos(theta)

F dot D



unit vector a/||a||



is a x b orthogonal or parallel to and and b orthogonal



||a x b|| (find angle between vectors) ||a||||b|| sin(theta)



area of parallelogram ||a x b||



Scalar triple product ||a dot (b x c)||



Parametric equa on with pt. (x0, y0, z0) vector (a, b, c) x = x0+at

y = y0+bt

z = z0+ct



Scalar equa on of plane with pt. (x0, y0, z0) vector (a, b, c) a(x-x0) + b (y-y0) + c(z-z0) = 0



Arc Length integral of ||r'(t)|| dt



Curvature (K(t)) (||r'(t) x r''(t) ||/||r'(t)||^3)