Calc 3 Chapter 15 Question and answers
already passed 2024
Conditions of using The Fundamental Theorem of Line Integrals - F is
conservative and f is a potential function
How to use the Fundamental Theorem of Line Integrals - ∫F * dr = f(endpt of
C) - f(startpt of C)
Line Integral of a Function - ∫ f ds = ∫ f(x(t),y(t),z(t))||r'(t)||dt
1 - parametrization the curve
2 - take derivative of parametrization
3 - find magnitude of derivative of parametrization
4 - substitute x(t) into x of f, y(t) into y of f, z(t) into z of f, multiply all by
magnitude of derivative of parametrization
5 - integrate
Conditions of using Green's Theorem - C is 2D and is the edge of R with
induced orientation matching the right-hand rule.
How to use Green's Theorem - ∫_c M dx + N dy = ∫∫_R N_x - M_y dA
Ways to do Surface Integral of a Vector Field - 2 ways - divergence theorem
or long way
How to do a Surface Integral of a Vector Field (w/o divergence) - ∫∫F·n dS = +-
∫∫_R F(x(u,v),y(u,v),z(u,v))·[r_u x r_v] dA
already passed 2024
Conditions of using The Fundamental Theorem of Line Integrals - F is
conservative and f is a potential function
How to use the Fundamental Theorem of Line Integrals - ∫F * dr = f(endpt of
C) - f(startpt of C)
Line Integral of a Function - ∫ f ds = ∫ f(x(t),y(t),z(t))||r'(t)||dt
1 - parametrization the curve
2 - take derivative of parametrization
3 - find magnitude of derivative of parametrization
4 - substitute x(t) into x of f, y(t) into y of f, z(t) into z of f, multiply all by
magnitude of derivative of parametrization
5 - integrate
Conditions of using Green's Theorem - C is 2D and is the edge of R with
induced orientation matching the right-hand rule.
How to use Green's Theorem - ∫_c M dx + N dy = ∫∫_R N_x - M_y dA
Ways to do Surface Integral of a Vector Field - 2 ways - divergence theorem
or long way
How to do a Surface Integral of a Vector Field (w/o divergence) - ∫∫F·n dS = +-
∫∫_R F(x(u,v),y(u,v),z(u,v))·[r_u x r_v] dA
