VECTOR CALCULUS FOR ADVANCED ENGINEERING MATHEMATICS

vector calculus

1. VECTOR CALCULUS BY: MARC JOSHUA PRUDENTE

2. VECTOR CALCULUS Vector Calculus is a branch of mathematics that extends ordinary calculus to functions of multiple variables and vector-valued functions. It focuses on vector fields, which assign a vector (having both magnitude and direction) to every point in space. 2

3. VECTOR CALCULUS It studies: Differentiation and Integration of vector fields. How vectors change, flow, or accumulate regions in space. At every point, there’s an arrow showing how fast and in which direction things move, called a vector field . 3

4. VECTOR FIELD, DIVERGENCE AND CURL A vector field is basically a way to assign a direction and magnitude (strength) to every point in space. Vector – an arrow with a length (how strong) and direction (which way it points) Field – defined all over space (or some region) Divergence and curl are properties of vector fields. Divergence – spreading or compressing at a point . Curl – rotating or swirling a t a point 4 Figure 1. Windspeed

5. VECTOR FIELD A vector field is a function that assigns a vector to every point in a region of space. such that for every point , the value is an n-dimensional vector .   5

6. VECTOR FIELD Example in 2D: For : where: : x-component of the field : y-component of the field Thus, each point in the plane has a vector attached to it.   6

7. VECTOR FIELD Representation of figure 2: where: : distance from origin to the point in the plane. Thus, in polar form: where: : angle the point makes with the axis.   7 Figure 2. Vector Field

8. VECTOR FIELD Substitute to the field: Compute magnitude: Get unit-direction vector by dividing by its magnitude : where: : components of unit vector pointing radially outward from the origin : same vector but mirrored across the y-axis. (x-component reversed)   8 Figure 2. Vector Field

9. VECTOR FIELD Thus: Near origin: vectors are short. Far from origin: vectors approach unit length 9 Figure 2. Vector Field

10. DIVERGENCE For a vector field , the divergence of F is: where: : divergence operator : components of vector field Thus: : field is expanding : field is contracting : field is incompressible   10

11. DIVERGENCE Let , Find the divergence of F. Solution: Components of vector field: Compute partial derivatives:   11 Add all components to get the divergence of F:  

12. DIVERGENCE Let , Find the divergence of F. Solution: Components of vector field: Compute partial derivatives:   12 Add all components to get the divergence of F:  

13. CURL For a vector field , the curl of F is: where: : curl : components of vector field   13

14. CURL Let , Find the curl of F. Solution: Components of vector field: Compute each component cross product: -component: -component: -component:   14 Add all components to get the curl of F:  

15. CURL Let , Find the curl of F. Solution: Components of vector field: Compute each component cross product: -component:   15 Add all components to get the curl of F:  

16. LINE INTEGRAL also called a path integral, is a way to integrate a vector field along a curve or path. It measures how much the vector field “flows along” that path. or where: : vector field :curve or path, described by position vector for . : vector differential, direction of curve and its magnitude   16 Figure 3. Approximating area under a curve

17. LINE INTEGRAL Compute the line integral of along the curve in the x-y plane for . Solution: Parameterize the curve: curve is , use x as parameter:   17 Differential vector then convert to differential scalar: Express in terms of :  

18. LINE INTEGRAL Substitute to integral then compute: Line integral is equal to 14.23   18

19. LINE INTEGRAL Compute the line integral of where C is the line segment from to in the x-y plane for . Solution: Parameterize the line segment using linear interpolation:   19 Differential vector then convert to differential scalar: Substitute to integral: Line integral is equal to 564.21.  

20. Fundamental theorem of line integrals Meaning/conditions: is a conservative (gradient) vector field. The integral depends only on the endpoints A and B, not on the path chosen between them. This is a direct analogue of the Fundamental Theorem of Calculus; it follows from the chain rule.   20

21. Fundamental theorem of line integrals 21 Find a function such that where C is any curve from to Solution: Get the vector field: Goal: Find one single function that fits both the conditions. Integrate the vector field with respect to x: Differentiate the vector field result with respect to y:   Compare the differential vector field result with the : Solve for : Substitute to the .  

22. Fundamental theorem of line integrals 22 Substitute the coordinates to the :  

23. GREEN’S THEOREM The circulation of a vector field around the boundary of a region equals the total ‘rotation’ (or curl) of the field inside that region. If C is a positively oriented, simple closed curve (counterclockwise direction),and D is the region enclosed by C, then: where: and : components of vector field. Left side: line integral around the boundary C Right side: double integral over the area D   23

24. GREEN’S THEOREM Use Green’s theorem to evaluate: where C is the triangle with vertices , , and . Solution: Identify P and Q:   24 Find the partial derivatives: Substitute to Green’s Theorem:  

25. GREEN’S THEOREM Describe region D (counterclockwise). Therefore:   25 Combine limits to double integral: Integrate with respect to y first since : Then integrate remaining:  

26. Vector functions for surfaces A parametric surface in is given by a vector function: where: : tangent vectors   26 Figure 4. Tracing a Surface

27. Vector functions for surfaces where: : tangent vectors : normal vector - unnormalized (perpendicular to the surface) : surface area element. : Flux   27 Figure 5. surface integral with vector field

28. Vector functions for surfaces Find the Flux for the paraboloid over the disk : Solution: Since vector field is not defined we can say that Compute the partial derivatives of the paraboloid. Convert disk to polar form to get the coordinates:   28  

29. Vector functions for surfaces Substitute into the formula of Flux. Since we are using theta as a limit, we need to convert back from rectangular to polar. Include Jacobian scaling factor for :   29 The Flux is .  

30. STOKE’S THEOREM where: : vector field :curve or path, described by position vector for . : vector differential, direction of curve and its magnitude   30 Figure 6. Vector Permeating a Surface

31. STOKE’S THEOREM If we have a vector field and C is given as . Find : Solution: Solve for the partial derivatives of the equation: Solve for curl F: -component: -component: -component:   31 Substitute the values into the Stoke’s equation: The answer is  

32. DIVERGENCE THEOREM   32 Figure 6. Vector Permeating a Surface

33. DIVERGENCE THEOREM If we have a vector field and C is given as z as coordinates given as . Find : Solution: Solve for the divergence: Substitute the values into the Gaussian equation:   33 The answer is  

34. Thank you