Green's theorem practice problems
Trade a line integral around a closed curve for a double integral inside it. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 1] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 3] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 2] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 2] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 1] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 2] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 1] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 3] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 1] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 1] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 2] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 1] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 1] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 1] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 1] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 1] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 3] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 1] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 2] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 1] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 3] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 3] \times [0, 3] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (y)\,dx + (3 x)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 3] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (x^{2} - y)\,dx + (x + y^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 3] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- y^{2})\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 1] \times [0, 2] \), counterclockwise.
Use Green's theorem to evaluate \( \displaystyle \oint_C (- x y)\,dx + (x y)\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 3] \), counterclockwise.