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4. ] Boundless.com/calculus?campaign_content=book_2286_ chapter_13&campaign_term=Calculus&utm_campaign=po werpoint&utm_medium=direct&utm_source=boundless Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus: Vector Calculus Vector Fields Line Integrals Conservative Vector Fields Green's Theorem Free to share, print, make copies and changes. Get yours at www.boundless.com Parametric Surfaces and Surface Integrals

5. ] Boundless.com/calculus?campaign_content=book_2286_ chapter_13&campaign_term=Calculus&utm_campaign=po werpoint&utm_medium=direct&utm_source=boundless Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus: Vector Calculus (continued) Surface Integrals of Vector Fields Divergence Stokes' Theorem Free to share, print, make copies and changes. Get yours at www.boundless.com

6. Vector Fields Advanced Topics in Single-Variable Calculus and... > Vector Fields Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium= direct&utm_source=boundless

7. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields can be constructed out of scalar fields using the gradient operator. Vector fields can be thought to represent the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (the rate of change of volume of a flow) and curl (the rotation of a flow). Vector Fields Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/alternative-to-essential-calculus-2nd-james-stewart-1133112293-9781133112297/advanced-topics-in- single-variable-calculus-and-an-introduction-to-multivariable-calculus-vector-calculus-13/vector-fields-101/vector-fields-225- 2882?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium=direct&utm_source=bou ndless Fig 1 View on Boundless.com Advanced Topics in Single-Variable Calculus and... > Vector Fields

8. Line Integrals Advanced Topics in Single-Variable Calculus and... > Line Integrals Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium= direct&utm_source=boundless

9. The value of the line integral is the sum of the values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). Many simple formulae in physics (for example, W=F·s) have natural continuous analogs in terms of line integrals (W=∫C F· ds). The line integral finds the work done on an object moving through an electric or gravitational field, for example. In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given field along a given curve. Line Integrals Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/alternative-to-essential-calculus-2nd-james-stewart-1133112293-9781133112297/advanced-topics-in- single-variable-calculus-and-an-introduction-to-multivariable-calculus-vector-calculus-13/line-integrals-102/line-integrals-226- 2884?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium=direct&utm_source=bou ndless Line Integral Over Scalar Field View on Boundless.com Advanced Topics in Single-Variable Calculus and... > Line Integrals

10. Conservative Vector Fields Advanced Topics in Single-Variable Calculus and... > Conservative Vector Fields Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium= direct&utm_source=boundless

11. Conservative vector fields have the following property: The line integral from one point to another is independent of the choice of path connecting the two points; it is path-independent. Conservative vector fields are also irrotational, meaning that (in three dimensions) they have vanishing curl. Conservative Vector Fields Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/alternative-to-essential-calculus-2nd-james-stewart-1133112293-9781133112297/advanced-topics-in- single-variable-calculus-and-an-introduction-to-multivariable-calculus-vector-calculus-13/conservative-vector-fields-103/conservative-vector-fields- 227- 2883?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium=direct&utm_source=bou ndless Fig 1 View on Boundless.com Advanced Topics in Single-Variable Calculus and... > Conservative Vector Fields

12. A vector field v is said to be conservative if there exists a scalar field [Equation 1] such that [Equation 2]. Advanced Topics in Single-Variable Calculus and... > Conservative Vector Fields Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/alternative-to-essential-calculus-2nd-james-stewart-1133112293-9781133112297/advanced-topics-in- single-variable-calculus-and-an-introduction-to-multivariable-calculus-vector-calculus-13/conservative-vector-fields-103/conservative-vector-fields- 227- 2883?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium=direct&utm_source=bou ndless Equation 1 View on Boundless.com Equation 2 View on Boundless.com

13. Green's Theorem Advanced Topics in Single-Variable Calculus and... > Green's Theorem Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium= direct&utm_source=boundless

14. Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the xy-plane. Considering only two-dimensional vector fields, Green's theorem is equivalent to the two-dimensional version of the divergence theorem. Green's theorem can be used to compute area by line integral. Green's Theorem Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/alternative-to-essential-calculus-2nd-james-stewart-1133112293-9781133112297/advanced-topics-in- single-variable-calculus-and-an-introduction-to-multivariable-calculus-vector-calculus-13/green-s-theorem-104/green-s-theorem-228- 2886?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium=direct&utm_source=bou ndless Computing area by line integral View on Boundless.com Advanced Topics in Single-Variable Calculus and... > Green's Theorem

15. Parametric Surfaces and Surface Integrals Advanced Topics in Single-Variable Calculus and... > Parametric Surfaces and Surface Integrals Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium= direct&utm_source=boundless

16. Parametric representation is the most general way to specify a surface. The curvature and arc length of curves on the surface can both be computed from a given parametrization. The same surface admits many different parametrizations. A surface integral is a definite integral taken over a surface. It can be thought of as the double integral analog of the line integral. Parametric Surfaces and Surface Integrals Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/alternative-to-essential-calculus-2nd-james-stewart-1133112293-9781133112297/advanced-topics-in- single-variable-calculus-and-an-introduction-to-multivariable-calculus-vector-calculus-13/parametric-surfaces-and-surface-integrals- 105/parametric-surfaces-and-surface-integrals-229- 2888?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium=direct&utm_source=bou ndless Kelvin-Stokes' Theorem View on Boundless.com Advanced Topics in Single-Variable Calculus and... > Parametric Surfaces and Surface Integrals

17. Surface Integrals of Vector Fields Advanced Topics in Single-Variable Calculus and... > Surface Integrals of Vector Fields Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium= direct&utm_source=boundless

18. The flux is defined as the quantity of fluid flowing through S in unit amount of time. To find the flux, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, and integrate the obtained field. Surface Integrals of Vector Fields Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/alternative-to-essential-calculus-2nd-james-stewart-1133112293-9781133112297/advanced-topics-in- single-variable-calculus-and-an-introduction-to-multivariable-calculus-vector-calculus-13/surface-integrals-of-vector-fields-106/surface-integrals-of- vector-fields-230- 2889?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium=direct&utm_source=bou ndless Kelvin-Stokes' Theorem View on Boundless.com Advanced Topics in Single-Variable Calculus and... > Surface Integrals of Vector Fields

19. This is expressed as [Equation 3]. Advanced Topics in Single-Variable Calculus and... > Surface Integrals of Vector Fields Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/alternative-to-essential-calculus-2nd-james-stewart-1133112293-9781133112297/advanced-topics-in- single-variable-calculus-and-an-introduction-to-multivariable-calculus-vector-calculus-13/surface-integrals-of-vector-fields-106/surface-integrals-of- vector-fields-230- 2889?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium=direct&utm_source=bou ndless Equation 3 View on Boundless.com

20. Curl and Divergence The Divergence Theorem Divergence Advanced Topics in Single-Variable Calculus and... > Divergence Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium= direct&utm_source=boundless

21. The curl is a vector operator that describes the infinitesimal rotation of a three- dimensional vector field. The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation. Divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point in terms of a signed scalar. Curl and Divergence Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/alternative-to-essential-calculus-2nd-james-stewart-1133112293-9781133112297/advanced-topics-in- single-variable-calculus-and-an-introduction-to-multivariable-calculus-vector-calculus-13/divergence-107/curl-and-divergence-231- 2887?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium=direct&utm_source=bou ndless Four Most Important Differential Operators View on Boundless.com Advanced Topics in Single-Variable Calculus and... > Divergence

22. The divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. In physics and engineering, the divergence theorem is usually applied in three dimensions. However, it generalizes to any number of dimensions. The Divergence Theorem Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/alternative-to-essential-calculus-2nd-james-stewart-1133112293-9781133112297/advanced-topics-in- single-variable-calculus-and-an-introduction-to-multivariable-calculus-vector-calculus-13/divergence-107/the-divergence-theorem-232- 2891?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium=direct&utm_source=bou ndless The Divergence Theorem View on Boundless.com Advanced Topics in Single-Variable Calculus and... > Divergence

23. Applying the divergence theorem, we can check that the equation [Equation 4] is nothing but an equation describing Coulomb force written in a differential form. Advanced Topics in Single-Variable Calculus and... > Divergence Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/alternative-to-essential-calculus-2nd-james-stewart-1133112293-9781133112297/advanced-topics-in- single-variable-calculus-and-an-introduction-to-multivariable-calculus-vector-calculus-13/divergence-107/the-divergence-theorem-232- 2891?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium=direct&utm_source=bou ndless Equation 4 View on Boundless.com

24. Stokes' Theorem Advanced Topics in Single-Variable Calculus and... > Stokes' Theorem Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium= direct&utm_source=boundless

25. The generalized Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω. Given a vector field, the Kelvin-Stokes theorem relates the integral of the curl of the vector field over some surface to the line integral of the vector field around the boundary of the surface. The Kelvin–Stokes theorem is a special case of the generalized Stokes' theorem. By applying the Stokes' theorem, you can show that the work done by electric field is path-independent. Stokes' Theorem Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/alternative-to-essential-calculus-2nd-james-stewart-1133112293-9781133112297/advanced-topics-in- single-variable-calculus-and-an-introduction-to-multivariable-calculus-vector-calculus-13/stokes-theorem-108/stokes-theorem-233- 2890?campaign_content=book_2286_chapter_13&campaign_term=Calculus&utm_campaign=powerpoint&utm_medium=direct&utm_source=bou ndless Kelvin-Stokes' Theorem View on Boundless.com Advanced Topics in Single-Variable Calculus and... > Stokes' Theorem

26. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

27. Key terms bijective both injective and surjective curl the vector field denoting the rotationality of a given vector field divergence a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar double integral An integral extended to functions of more than one real variable electric potential the potential energy per unit charge at a point in a static electric field; voltage flux the rate of transfer of energy (or another physical quantity) through a given surface, specifically electric flux, magnetic flux gradient of a function y = f(x) or the graph of such a function, the rate of change of y with respect to x; that is, the amount by which y changes for a certain (often unit) change in x Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and...

28. gradient of a function y = f(x) or the graph of such a function, the rate of change of y with respect to x; that is, the amount by which y changes for a certain (often unit) change in x line integral An integral the domain of whose integrand is a curve. parametrization Is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object. vector field a construction in which each point in a Euclidean space is associated with a vector; a function whose range is a vector space Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and...

29. Fig 1 The above field v(x,y,z) = (−y/(x2+y2), +x/(x2+y2), 0) includes a vortex at its center, meaning it is non-irrotational; it is neither conservative, nor does it have path independence. However, any simply connected subset that excludes the vortex line (0,0,z) will have zero curl, ∇ v = 0. Such vortex-free regions are examples of irrotational vector fields. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Conservative vector field." CC BY http://en.wikipedia.org/wiki/Conservative_vector_field View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Conservative_vector_fieldView on Boundless.com Advanced Topics in Single-Variable Calculus and...

30. Line Integral Over Scalar Field The line integral over a scalar field f can be thought of as the area under the curve C along a surface z = f(x,y), described by the field. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Line integral." CC BY http://en.wikipedia.org/wiki/Line_integral View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Line_integralView on Boundless.com Advanced Topics in Single-Variable Calculus and...

31. Kelvin-Stokes' Theorem An illustration of the Kelvin–Stokes theorem, with surface Σ, its boundary ∂, and the "normal" vector n. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Stokes' theorem." CC BY http://en.wikipedia.org/wiki/Stokes%2527_theorem View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Stokes%2527_theoremView on Boundless.com Advanced Topics in Single-Variable Calculus and...

32. The Divergence Theorem The divergence theorem can be used to calculate a flux through a closed surface that fully encloses a volume, like any of the surfaces on the left. It can not directly be used to calculate the flux through surfaces with boundaries, like those on the right. (Surfaces are blue, boundaries are red. ) Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Divergence theorem." CC BY http://en.wikipedia.org/wiki/Divergence_theorem View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Divergence_theoremView on Boundless.com Advanced Topics in Single-Variable Calculus and...

33. Four Most Important Differential Operators Gradient, curl, divergence, and Laplacian are four most important differential operators. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Vector calculus." CC BY http://en.wikipedia.org/wiki/Vector_calculus View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Vector_calculusView on Boundless.com Advanced Topics in Single-Variable Calculus and...

34. Computing area by line integral D is a simple region with its boundary consisting of the curves C1, C2, C3, C4. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia. CC BY-SA http://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Green%2527s-theorem-simple-region.svg/429px-Green%2527s-theorem-simple- region.svg.png View on Boundless.comCC BY-SAhttp://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Green%2527s-theorem-simple-region.svg/429px-Green%2527s-theorem-simple- region.svg.pngView on Boundless.com Advanced Topics in Single-Variable Calculus and...

35. Line Integral Over Scalar Field The line integral over a scalar field f can be thought of as the area under the curve C along a surface z = f(x,y), described by the field. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Line integral." CC BY http://en.wikipedia.org/wiki/Line_integral View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Line_integralView on Boundless.com Advanced Topics in Single-Variable Calculus and...

36. Kelvin-Stokes' Theorem An illustration of the Kelvin–Stokes theorem, with surface Σ, its boundary ∂, and the "normal" vector n. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Stokes' theorem." CC BY http://en.wikipedia.org/wiki/Stokes%2527_theorem View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Stokes%2527_theoremView on Boundless.com Advanced Topics in Single-Variable Calculus and...

37. Kelvin-Stokes' Theorem An illustration of the Kelvin–Stokes theorem, with surface Σ, its boundary ∂, and the "normal" vector n. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Stokes' theorem." CC BY http://en.wikipedia.org/wiki/Stokes%2527_theorem View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Stokes%2527_theoremView on Boundless.com Advanced Topics in Single-Variable Calculus and...

38. Fig 1 Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Green's theorem." CC BY http://en.wikipedia.org/wiki/Green%2527s_theorem View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Green%2527s_theoremView on Boundless.com Advanced Topics in Single-Variable Calculus and...

39. Fig 1 Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Vector field." CC BY http://en.wikipedia.org/wiki/Vector_field#Examples View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Vector_field#ExamplesView on Boundless.com Advanced Topics in Single-Variable Calculus and...

40. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... Vector fields can be constructed A) by assigning a vector to each point in a subset of Hilbert space B) Vector fields can be constructed out of scalar fields using the gradient operator C) by assigning a vector to each point in a subset of pseudo-Euclidean space D) out of vector fields using the tensor operator

41. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... Vector fields can be constructed A) by assigning a vector to each point in a subset of Hilbert space B) Vector fields can be constructed out of scalar fields using the gradient operator C) by assigning a vector to each point in a subset of pseudo-Euclidean space D) out of vector fields using the tensor operator

42. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... The value of the line integral is the A) unweighted sum of the values of the field at all points on the curve B) unweighted product of the values of the field at all points on the curve C) sum of the values of the field at all points on the curve weighted by scalar function on the curve D) sum of the values of the field at all points on the curve weighted by vector function on the curve

43. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... The value of the line integral is the A) unweighted sum of the values of the field at all points on the curve B) unweighted product of the values of the field at all points on the curve C) sum of the values of the field at all points on the curve weighted by scalar function on the curve D) sum of the values of the field at all points on the curve weighted by vector function on the curve

44. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... A line integral is an integral where the function to be integrated is evaluated A) over a 3-dimensional domain B) along a line C) over a surface D) in a point

45. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... A line integral is an integral where the function to be integrated is evaluated A) over a 3-dimensional domain B) along a line C) over a surface D) in a point

46. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... Conservative vector fields A) are path dependent and irrational B) are path dependent and rational C) are path independent and irrational D) are path independent and rational

47. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... Conservative vector fields A) are path dependent and irrational B) are path dependent and rational C) are path independent and irrational D) are path independent and rational

48. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... A surface integral is an integral where the function to be integrated is evaluated A) along a line B) over a 3-dimensional domain C) in a point D) over a surface

49. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... A surface integral is an integral where the function to be integrated is evaluated A) along a line B) over a 3-dimensional domain C) in a point D) over a surface

50. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... A parametric surface is a surface A) in the Euclidean space which is defined by a parametric equation with two parameters B) in the non-Euclidean space which is defined by a parametric equation with two parameters C) in the non-Euclidean space which is defined by a parametric equation with three parameters D) in the Euclidean space which is defined by a parametric equation with three parameters

51. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... A parametric surface is a surface A) in the Euclidean space which is defined by a parametric equation with two parameters B) in the non-Euclidean space which is defined by a parametric equation with two parameters C) in the non-Euclidean space which is defined by a parametric equation with three parameters D) in the Euclidean space which is defined by a parametric equation with three parameters

52. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... Surface integral of a scalar field A) can not be used to define the surface integral of vector fields B) can be used to define the surface integral of vector fields with the result being a scalar C) can be used to define the surface integral of vector fields with the result being a vector D) can not be used to define the surface integral of vector fields since the result will be a scalar

53. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... Surface integral of a scalar field A) can not be used to define the surface integral of vector fields B) can be used to define the surface integral of vector fields with the result being a scalar C) can be used to define the surface integral of vector fields with the result being a vector D) can not be used to define the surface integral of vector fields since the result will be a scalar

54. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... The curl is a vector operator that A) describes the infinitesimal rotation of a three-dimensional vector field B) measures the magnitude of a vector field's source at a given point in terms of a signed scalar C) points in the direction of the greatest rate of increase of the scalar field D) is given by the divergence of the gradient of a function on Euclidean space

55. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... The curl is a vector operator that A) describes the infinitesimal rotation of a three-dimensional vector field B) measures the magnitude of a vector field's source at a given point in terms of a signed scalar C) points in the direction of the greatest rate of increase of the scalar field D) is given by the divergence of the gradient of a function on Euclidean space

56. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... The direction of the curl is A) perpendicular to the axis of rotation, as determined by the right-hand rule B) parallel to the axis of rotation, as determined by the left-leg rule C) perpendicular to the axis of rotation, as determined by the left-leg rule D) parallel to the axis of rotation, as determined by the right-hand rule

57. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... The direction of the curl is A) perpendicular to the axis of rotation, as determined by the right-hand rule B) parallel to the axis of rotation, as determined by the left-leg rule C) perpendicular to the axis of rotation, as determined by the left-leg rule D) parallel to the axis of rotation, as determined by the right-hand rule

58. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... The magnitude of the curl is A) smaller than to the magnitude of rotation B) larger than to the magnitude of rotation C) equal to the magnitude of rotation D) not equal to the magnitude of rotation

59. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... The magnitude of the curl is A) smaller than to the magnitude of rotation B) larger than to the magnitude of rotation C) equal to the magnitude of rotation D) not equal to the magnitude of rotation

60. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... The divergence is a vector operator that A) describes the infinitesimal rotation of a three-dimensional vector field B) points in the direction of the greatest rate of increase of the scalar field C) measures the magnitude of a vector field's source at a given point in terms of a signed scalar D) is given by the divergence of the gradient of a function on Euclidean space

61. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... The divergence is a vector operator that A) describes the infinitesimal rotation of a three-dimensional vector field B) points in the direction of the greatest rate of increase of the scalar field C) measures the magnitude of a vector field's source at a given point in terms of a signed scalar D) is given by the divergence of the gradient of a function on Euclidean space

62. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... The divergence theorem relates flow of a vector field through a surface to the A) line integral of the field around the boundary B) behavior of the vector field outside the surface C) flow of a vector field through a surface to the behavior of the vector field inside the surface D) double integral of the field around the boundary

63. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... The divergence theorem relates flow of a vector field through a surface to the A) line integral of the field around the boundary B) behavior of the vector field outside the surface C) flow of a vector field through a surface to the behavior of the vector field inside the surface D) double integral of the field around the boundary

64. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... According to the divergence theorem, the outward flux of a vector field through a closed surface is A) larger than the volume integral of the divergence over the region inside the surface B) equal to the volume integral of the divergence over the region inside the surface C) smaller than the volume integral of the divergence over the region inside the surface D) not equal to the volume integral of the divergence over the region inside the surface

65. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... According to the divergence theorem, the outward flux of a vector field through a closed surface is A) larger than the volume integral of the divergence over the region inside the surface B) equal to the volume integral of the divergence over the region inside the surface C) smaller than the volume integral of the divergence over the region inside the surface D) not equal to the volume integral of the divergence over the region inside the surface

66. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... The Stokes' theorem relates the integral of the curl of a vector field over a surface to the A) behavior of the vector field inside the surface B) double integral of the field around the boundary C) line integral of the field around the boundary D) behavior of the vector field outside the surface

67. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... The Stokes' theorem relates the integral of the curl of a vector field over a surface to the A) behavior of the vector field inside the surface B) double integral of the field around the boundary C) line integral of the field around the boundary D) behavior of the vector field outside the surface

68. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... By applying the Stokes' theorem, we can show that the work done by electric field is A) path-dependent B) positive C) path-independent D) negative

69. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... By applying the Stokes' theorem, we can show that the work done by electric field is A) path-dependent B) positive C) path-independent D) negative

70. Attribution Wikipedia. "Line integral." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Line_integralCC BY-SA 3.0http://en.wikipedia.org/wiki/Line_integral Wiktionary. "vector field." CC BY-SA 3.0 http://en.wiktionary.org/wiki/vector+fieldCC BY-SA 3.0http://en.wiktionary.org/wiki/vector+field Wiktionary. "bijective." CC BY-SA 3.0 http://en.wiktionary.org/wiki/bijectiveCC BY-SA 3.0http://en.wiktionary.org/wiki/bijective Wikipedia. "Line integral." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Line_integralCC BY-SA 3.0http://en.wikipedia.org/wiki/Line_integral Wiktionary. "vector field." CC BY-SA 3.0 http://en.wiktionary.org/wiki/vector+fieldCC BY-SA 3.0http://en.wiktionary.org/wiki/vector+field Wiktionary. "bijective." CC BY-SA 3.0 http://en.wiktionary.org/wiki/bijectiveCC BY-SA 3.0http://en.wiktionary.org/wiki/bijective Wikipedia. "Line integral." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Line_integralCC BY-SA 3.0http://en.wikipedia.org/wiki/Line_integral Wiktionary. "vector field." CC BY-SA 3.0 http://en.wiktionary.org/wiki/vector+fieldCC BY-SA 3.0http://en.wiktionary.org/wiki/vector+field Wiktionary. "bijective." CC BY-SA 3.0 http://en.wiktionary.org/wiki/bijectiveCC BY-SA 3.0http://en.wiktionary.org/wiki/bijective Wikipedia. "Green's theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Green's_theoremCC BY-SA 3.0http://en.wikipedia.org/wiki/Green's_theorem Wikipedia. "Green's theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Green's_theoremCC BY-SA 3.0http://en.wikipedia.org/wiki/Green's_theorem Wiktionary. "line integral." CC BY-SA 3.0 http://en.wiktionary.org/wiki/line+integralCC BY-SA 3.0http://en.wiktionary.org/wiki/line+integral Wikipedia. "double integral." CC BY-SA 3.0 http://en.wikipedia.org/wiki/double%20integralCC BY-SA 3.0http://en.wikipedia.org/wiki/double%20integral Wikipedia. "Divergence." CC BY-SA 3.0 http://en.wikipedia.org/wiki/DivergenceCC BY-SA 3.0http://en.wikipedia.org/wiki/Divergence Wikipedia. "Curl (mathematics)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Curl_%2528mathematics%2529CC BY-SA 3.0http://en.wikipedia.org/wiki/Curl_%2528mathematics%2529 Wikipedia. "Vector calculus." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Vector_calculusCC BY-SA 3.0http://en.wikipedia.org/wiki/Vector_calculus Wiktionary. "gradient." CC BY-SA 3.0 http://en.wiktionary.org/wiki/gradientCC BY-SA 3.0http://en.wiktionary.org/wiki/gradient Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and...

71. Wikipedia. "Kelvin–Stokes theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Kelvin%2525E2%252580%252593Stokes_theoremCC BY-SA 3.0 http://en.wikipedia.org/wiki/Kelvin%2525E2%252580%252593Stokes_theorem Wikipedia. "Stokes' theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Stokes%252527_theoremCC BY-SA 3.0http://en.wikipedia.org/wiki/Stokes%252527_theorem Wiktionary. "gradient." CC BY-SA 3.0 http://en.wiktionary.org/wiki/gradientCC BY-SA 3.0http://en.wiktionary.org/wiki/gradient Wiktionary. "electric potential." CC BY-SA 3.0 http://en.wiktionary.org/wiki/electric+potentialCC BY-SA 3.0http://en.wiktionary.org/wiki/electric+potential Wiktionary. "curl." CC BY-SA 3.0 http://en.wiktionary.org/wiki/curlCC BY-SA 3.0http://en.wiktionary.org/wiki/curl Wikipedia. "Stokes' theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Stokes%2527_theoremCC BY-SA 3.0http://en.wikipedia.org/wiki/Stokes%2527_theorem Wikipedia. "Kelvin–Stokes theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Kelvin%25E2%2580%2593Stokes_theoremCC BY-SA 3.0http://en.wikipedia.org/wiki/Kelvin%25E2%2580%2593Stokes_theorem Wiktionary. "vector field." CC BY-SA 3.0 http://en.wiktionary.org/wiki/vector+fieldCC BY-SA 3.0http://en.wiktionary.org/wiki/vector+field Wikipedia. "parametrization." CC BY-SA 3.0 http://en.wikipedia.org/wiki/parametrizationCC BY-SA 3.0http://en.wikipedia.org/wiki/parametrization Wiktionary. "flux." CC BY-SA 3.0 http://en.wiktionary.org/wiki/fluxCC BY-SA 3.0http://en.wiktionary.org/wiki/flux Wikipedia. "Stokes' theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Stokes%2527_theoremCC BY-SA 3.0http://en.wikipedia.org/wiki/Stokes%2527_theorem Wikipedia. "Kelvin–Stokes theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Kelvin%25E2%2580%2593Stokes_theoremCC BY-SA 3.0http://en.wikipedia.org/wiki/Kelvin%25E2%2580%2593Stokes_theorem Wiktionary. "gradient." CC BY-SA 3.0 http://en.wiktionary.org/wiki/gradientCC BY-SA 3.0http://en.wiktionary.org/wiki/gradient Wiktionary. "electric potential." CC BY-SA 3.0 http://en.wiktionary.org/wiki/electric+potentialCC BY-SA 3.0http://en.wiktionary.org/wiki/electric+potential Wiktionary. "curl." CC BY-SA 3.0 http://en.wiktionary.org/wiki/curlCC BY-SA 3.0http://en.wiktionary.org/wiki/curl Wikipedia. "Divergence theorem." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Divergence_theoremCC BY-SA 3.0http://en.wikipedia.org/wiki/Divergence_theorem Wiktionary. "flux." CC BY-SA 3.0 http://en.wiktionary.org/wiki/fluxCC BY-SA 3.0http://en.wiktionary.org/wiki/flux Wiktionary. "divergence." CC BY-SA 3.0 http://en.wiktionary.org/wiki/divergenceCC BY-SA 3.0http://en.wiktionary.org/wiki/divergence Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and...