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 vectors double integrals vector fields line integrals

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

17calculus > vector fields > green's theorem

 Theorem Relationships Green's Theorem Explanation Alternative Forms Meaning Proofs Practice

Relationships Between Green's Theorem, Stokes' Theorem and the Divergence Theorem

Green's Theorem has two forms, the circulation form and the divergence form. Green's Theorem is in two dimensions, While Stokes' Theorem is the three-dimensional form of the circulation form of Green's Theorem. Similarly, the Divergence Theorem is the three-dimensional form of the divergence form of Green's Theorem. The following table summarizes these relationships.

2-dim

3-dim

Green's Theorem (circulation form)

Stokes' Theorem

Greens' Theorem (divergence form)

Divergence Theorem

If you want a complete lecture on this topic, here is a video.

### Prof Leonard - Green's Theorem [1hr-45min-53secs]

video by Prof Leonard

Green's Theorem

IF WE HAVE

- a simply-connected region $$\mathcal{R}$$ which is bounded by a smooth curve $$\mathcal{C}$$, oriented anticlockwise (counter-clockwise) and traversed once, and
- a vector field $$\vec{F} = M\hat{i}+N\hat{j}$$ which is continuously differentiable on the region $$\mathcal{R}$$.

THEN THIS EQUATION HOLDS

$$\displaystyle{ \oint_{\mathcal{C}}{M~dx+N~dy} = \iint\limits_{\mathcal{R}}{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} dA } }$$

Notation - - The circle on the integral sign clarifies that the curve $$\mathcal{C}$$ is a closed curve. It is not necessary and not all books and instructors use it.

Green's Theorem relates the calculation of line integrals to double integrals over area in the special case when the line integral completely encloses a single region in the plane. This enclosed region must be simply-connected, i.e. the enclosing line must not cross itself and it must consist of one simple closed curve. This theorem is very powerful and allows us to evaluate line integrals using double integrals.

Let's get started with a video. This video is excellent in explaining Green's Theorem, including several examples. It covers the idea of simply-connected regions as well.

### Dr Chris Tisdell - What is Green's Theorem? [47mins-49secs]

The last example in this video has an incorrect answer. The final answer should be $$14/3$$.

video by Dr Chris Tisdell

Alternative Forms of Green's Theorem

Okay, so the theorem at the top of the page is the basic form of Green's Theorem. There are two other alternative forms that are equivalent to this basic form. These forms were touched on in the previous video.

1. circulation or curl ( in 3-dim called Stokes' Theorem )

$$\displaystyle{ \oint_{\mathcal{C}}{ \vec{F} \cdot \vec{T} ~ds} = }$$ $$\displaystyle{ \iint_{\mathcal{R}}{(curl ~\vec{F}) \cdot \hat{k} ~ dA} }$$

$$\displaystyle{ \oint_{\mathcal{C}}{\vec{F} \cdot d\vec{r} } = }$$ $$\displaystyle{ \iint_{\mathcal{R}}{ \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right) dA} }$$

2. flux or divergence ( in 3-dim called the Divergence Theorem )

$$\displaystyle{ \oint_{\mathcal{C}}{ \vec{F} \cdot \vec{n} ~ds} = }$$ $$\displaystyle{ \iint_{\mathcal{R}}{div~\vec{F}~dA} }$$

$$\displaystyle{ \iint_{\mathcal{R}}{ \left( \frac{\partial N}{\partial y} + \frac{\partial M}{\partial x}\right) dA} }$$

In the table above, $$\vec{T}$$ is the unit tangent vector and $$\vec{n}$$ is the outward unit normal vector.

This next video explains these alternative forms in more detail and has some examples.

### Dr Chris Tisdell - Green's Theorem [39mins-13secs]

video by Dr Chris Tisdell

Another by-product of Green's Theorem is that we can now use a line integral to calculate area. Here is the equation.

$$\displaystyle{ A = \frac{1}{2} \oint_{\mathcal{C}}{x~dy - y~dx} }$$

The previous video shows the derivation of this equation. If you skipped it, now is the time to go back and watch it.

Meaning of Green's Theorem

So, now that you know how to use Green's Theorem, what is really going on here? This next video gives a quick review of the Theorem and then explains the interpretation and meaning of it.

### Michael Hutchings - Multivariable calculus 4.4.1: Review and interpretation of Green's theorem [7mins-25secs]

Proofs of Green's Theorem

Here are several video proofs of Green's Theorem. Each instructor proves Green's Theorem differently. So it will help you to understand the theorem if you watch all of these videos.

### David Metzler - Greens Theorem Proof [16mins-22secs]

video by David Metzler

### Khan Academy - Green's Theorem Proof (2) [19mins-26secs]

Okay, you are now ready for some practice problems. Next, you need to understand how to parameterize surfaces before you go on to surface integrals and the three dimensional versions of Green's Theorem, Stokes' Theorem and the Divergence Theorem.

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Instructions - - Unless otherwise instructed, use Green's Theorem to evaluate these integrals.

Basic Problems

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ (y^2 ~dx + 3xy ~dy) } }$$ where $$\mathcal{C}$$ is the boundary of the semiannular region D in the upper half plane between the circles $$x^2 + y^2 = 1$$ and $$x^2 + y^2 = 4$$.

Problem Statement

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ (y^2 ~dx + 3xy ~dy) } }$$ where $$\mathcal{C}$$ is the boundary of the semiannular region D in the upper half plane between the circles $$x^2 + y^2 = 1$$ and $$x^2 + y^2 = 4$$.

Solution

### 1569 solution video

video by Dr Chris Tisdell

Calculate the outward flux of $$\vec{F} = -x\hat{i} + 2y\hat{j}$$ over the square with corners $$(\pm 1, \pm 1)$$ where the unit normal is outward-pointing.

Problem Statement

Calculate the outward flux of $$\vec{F} = -x\hat{i} + 2y\hat{j}$$ over the square with corners $$(\pm 1, \pm 1)$$ where the unit normal is outward-pointing.

Solution

### 1570 solution video

video by Dr Chris Tisdell

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ y^2~dx+x^2~dy } }$$ where $$\mathcal{C}$$ is the triangle bounded by the lines $$x=0, x+y=1, y=0$$

Problem Statement

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ y^2~dx+x^2~dy } }$$ where $$\mathcal{C}$$ is the triangle bounded by the lines $$x=0, x+y=1, y=0$$

Solution

### 1571 solution video

video by Dr Chris Tisdell

Let $$\mathcal{C}$$ be the circle $$x^2 + y^2 = 4.$$ Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ \left[ (3y-e^{\tan^{-1}x})~dx + (7x + \sqrt{y^4+1}) ~dy \right] } }$$

Problem Statement

Let $$\mathcal{C}$$ be the circle $$x^2 + y^2 = 4.$$ Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ \left[ (3y-e^{\tan^{-1}x})~dx + (7x + \sqrt{y^4+1}) ~dy \right] } }$$

Solution

### 1572 solution video

video by Dr Chris Tisdell

Evaluate $$\displaystyle{ \int_{\mathcal{C}}{ x~dx - x^2y^2~dy } }$$ where $$\mathcal{C}$$ is the triangle with vertices $$(0,0)$$, $$(0,1)$$ and $$(1,1)$$.

Problem Statement

Evaluate $$\displaystyle{ \int_{\mathcal{C}}{ x~dx - x^2y^2~dy } }$$ where $$\mathcal{C}$$ is the triangle with vertices $$(0,0)$$, $$(0,1)$$ and $$(1,1)$$.

Solution

### 1573 solution video

video by PatrickJMT

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ P~dx + Q~dy } }$$ where $$\mathcal{C}$$ is the square with corners $$(\pm 1, \pm 1)$$; $$P(x,y) = x + y^2$$ and $$Q(x,y) = y + x^2$$.

Problem Statement

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ P~dx + Q~dy } }$$ where $$\mathcal{C}$$ is the square with corners $$(\pm 1, \pm 1)$$; $$P(x,y) = x + y^2$$ and $$Q(x,y) = y + x^2$$.

Solution

### 1574 solution video

video by Krista King Math

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ P~dx+Q~dy } }$$ where $$\mathcal{C}$$ is the triangle with corners $$(0,0)$$, $$(1,1)$$, $$(2,0)$$, $$P(x,y)=y+e^x$$ and $$Q(x,y)=2x^2+\cos y$$.

Problem Statement

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ P~dx+Q~dy } }$$ where $$\mathcal{C}$$ is the triangle with corners $$(0,0)$$, $$(1,1)$$, $$(2,0)$$, $$P(x,y)=y+e^x$$ and $$Q(x,y)=2x^2+\cos y$$.

Solution

### 1575 solution video

video by Krista King Math

Evaluate $$\displaystyle{ \int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}} }$$ where $$\mathcal{C}$$ is the circle $$x^2+y^2=4$$ and $$\vec{F}=\langle -y, x \rangle$$.

Problem Statement

Evaluate $$\displaystyle{ \int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}} }$$ where $$\mathcal{C}$$ is the circle $$x^2+y^2=4$$ and $$\vec{F}=\langle -y, x \rangle$$.

Solution

### 1576 solution video

video by MIP4U

Evaluate $$\displaystyle{\int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}}}$$ where $$\mathcal{C}$$ is along $$y=x^2/2$$ and $$y=x$$ and $$\vec{F}=\langle y^2, x^2 \rangle$$.

Problem Statement

Evaluate $$\displaystyle{\int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}}}$$ where $$\mathcal{C}$$ is along $$y=x^2/2$$ and $$y=x$$ and $$\vec{F}=\langle y^2, x^2 \rangle$$.

Solution

### 1577 solution video

video by MIP4U

Evaluate $$\displaystyle{\int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}}}$$ where $$\vec{F}=\langle y^3, 6xy^2 \rangle$$ and $$\mathcal{C}$$ is the curve with the 3 segments:
1. $$y=0, 0 \leq x \leq 4$$, 2. $$x=4, 0 \leq y \leq 2$$, 3. $$y=\sqrt{x}, 0 \leq x \leq 4$$

Problem Statement

Evaluate $$\displaystyle{\int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}}}$$ where $$\vec{F}=\langle y^3, 6xy^2 \rangle$$ and $$\mathcal{C}$$ is the curve with the 3 segments:
1. $$y=0, 0 \leq x \leq 4$$, 2. $$x=4, 0 \leq y \leq 2$$, 3. $$y=\sqrt{x}, 0 \leq x \leq 4$$

Solution

### 1578 solution video

video by MIP4U

Evaluate $$\displaystyle{ \int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}} }$$ where $$\mathcal{C}$$ is boundary between the two circles $$x^2+y^2=1$$ and $$x^2+y^2=3$$ that lies in the first quadrant and $$\vec{F} = \langle e^x+2xy, 4x^2+\sin y \rangle$$.

Problem Statement

Evaluate $$\displaystyle{ \int_{\mathcal{C}}{\vec{F} \cdot d\vec{r}} }$$ where $$\mathcal{C}$$ is boundary between the two circles $$x^2+y^2=1$$ and $$x^2+y^2=3$$ that lies in the first quadrant and $$\vec{F} = \langle e^x+2xy, 4x^2+\sin y \rangle$$.

Solution

### 1579 solution video

video by MIP4U

Determine the flux of the field $$\vec{F} = \langle -x, -y \rangle$$ across $$x^2+y^2=16$$.

Problem Statement

Determine the flux of the field $$\vec{F} = \langle -x, -y \rangle$$ across $$x^2+y^2=16$$.

Solution

### 1580 solution video

video by MIP4U

Calculate the outward flux of $$\vec{F} = -x\hat{i} + 2y\hat{j}$$ over the square with corners $$(\pm 1, \pm 1)$$ where the unit normal is outward-pointing.

Problem Statement

Calculate the outward flux of $$\vec{F} = -x\hat{i} + 2y\hat{j}$$ over the square with corners $$(\pm 1, \pm 1)$$ where the unit normal is outward-pointing.

4

Problem Statement

Calculate the outward flux of $$\vec{F} = -x\hat{i} + 2y\hat{j}$$ over the square with corners $$(\pm 1, \pm 1)$$ where the unit normal is outward-pointing.

Solution

### 1984 solution video

video by Dr Chris Tisdell

4

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ 2xy~dx + (x^2+y^2)~dy } }$$ where $$\mathcal{C}$$ is $$4x^2 + 9y^2 = 36$$.

Problem Statement

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ 2xy~dx + (x^2+y^2)~dy } }$$ where $$\mathcal{C}$$ is $$4x^2 + 9y^2 = 36$$.

0

Problem Statement

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ 2xy~dx + (x^2+y^2)~dy } }$$ where $$\mathcal{C}$$ is $$4x^2 + 9y^2 = 36$$.

Solution

### 2200 solution video

video by Michel vanBiezen

0

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{(x^2-y^2)~dx + 2xy~dy} }$$ on the curve which bounds the region $$\mathcal{R}: \{(x,y): 0 \leq x \leq 1, 2x^2 \leq y \leq 2x \}$$

Problem Statement

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{(x^2-y^2)~dx + 2xy~dy} }$$ on the curve which bounds the region $$\mathcal{R}: \{(x,y): 0 \leq x \leq 1, 2x^2 \leq y \leq 2x \}$$

Solution

### 1581 solution video

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ 2y~dx-3x~dy } }$$ on the unit circle.

Problem Statement

Evaluate $$\displaystyle{ \oint_{\mathcal{C}}{ 2y~dx-3x~dy } }$$ on the unit circle.

Solution

### 1582 solution video

Intermediate Problems

Verify Green's theorem for the line integral $$\displaystyle{ \oint_{\mathcal{C}}{ xy~dx + x~dy } }$$ where $$\mathcal{C}$$ is the unit circle.

Problem Statement

Verify Green's theorem for the line integral $$\displaystyle{ \oint_{\mathcal{C}}{ xy~dx + x~dy } }$$ where $$\mathcal{C}$$ is the unit circle.

Solution

### 1568 solution video

video by Dr Chris Tisdell