yA 7/ャ ↑ア 一→ー ア

Q: Interpret the divergence of a vector field.

A:

Q: How do you graph the vector field F = ⟨ƒ(x, y), g(x, y)⟩?

A: Given: vector field

Q: Find a formula for the vector field (vectors are shown as line segments, with a dot at the tail of…

A: This question is about application of derivative

Q: Sketch the following gradient vector field, F(x, y, z) = 2xi - 2yj – 2xk. Then find the vector…

A: The given problem is to sketch the given vector field and also to find the gradient of the given…

Q: In words, what is a vector field? Give some physical examples of vector fields

A: Vector feild can be defined in 2 dimensions as well as 3 dimensions.A vector field F→ in ℝ2 is an…

Q: State the formula for Curl of a Conservative Vector Field.

A: Given, To State the formula for Curl of a Conservative Vector Field.

Q: Find llFll and sketch several representative vectors in the vector field.

A:

Q: A vector field in space is conservative. Is the vector field irrotational? Explain.

A: Query-  A vector field in space is conservative. Is the vector field irrotational? Explain. Answer-…

Q: What does it mean if the curl of a vector field is zero throughout a region?

A:

Q: Match the vector field with its graph. F(x, y) = yi 址 四 場 x Q 6 電

A: F(x,y)=yi So at any point vector magnitude is same as y-cooridnate Also, if y is negative [below the…

Q: Is the following vector field conservative? F =

A: Yes the vector field is conservative

Q: Consider the vector field F that is depicted below. On which of the parameterized curves is the…

A: Given the vector field F in the image

Q: Match the vector field with its graph. F(x, y) = yi KE y -5+

A:

Q: Find the curl of the vector field at the given point. F(x, y, z) = x2zi – 2xzj + yzk; (9, -9, 5)

A:

Q: Consider the vector field v(x)

A: Given, the vector field        v(x)=yx2+y2i^-xx2+y2j^+6k^                               ...(1)

Q: Match the vector field with its graph. F(x, y) = yi 5 5 -5+ サ

A: To match the given vector field to its graph, where given graphs are given below.

Q: Look at the vector field in the plane F. Xi - j

A: “Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: Determine the vector field of F. F(x, y) 3D yi — хј

A: Let us make a table to draw the vector field as shown in the image below.

Q: Sketch a few representative vectors of vector field F = ⟨0, 1⟩along the line y = 2.

A:

Q: Compute the curl of the vector field F⃗ =7zi⃗ +6yj⃗ +4xk⃗

A:

Q: Sketch the vector field in the plane along with its horizonta and Vertical a representative…

A:

Q: 9. Sketch the vector field F(x, y) : :(yi — х). Vx? + y?

A: Consider the given filed, Fx,y=1x2+y2yi−xj Further simplify. Fx,y=yx2+y2i-xx2+y2i=〈yx2+y2,-xx2+y2〉

Q: Describe the vector field by drawing some of its vectors. F(x, y) = 5xi - 5yj WebAssign Plot

A:

Q: Is there a vector field Ğ on R³ such that curl Ġ = (xyz,–y*z³,y*z*)?

A: the given function is, G→=xyzi^-y4z3j^+y3z4k^ If there exists a vector field G then the divergence…

Q: Make a sketch of the following vector field. F = (-x,2y) %3D

A: We sketch the vector field, we choose  the  points and sketch them.

Q: Give three equivalent properties of conservative vector fields.

A:

Q: Determine whether the vector field is conservative. F(x, y) = 6y6(yi + 7xj) conservative not…

A:

Q: Compute the flux of the vector field (x, (æ°, – ay*), out of the rectangle with vertices (0,0),…

A: To compute the flux of the vector field x3,-xy4, out of the rectanglewith vertices 0,0,2,0,2,4…

Q: Write a formula for a two-dimensional vector field which has all vectors of length 7 and…

A:

Q: Calculate the Divergence and curl of the vector field.

A: Solve for divergence and curl

Q: The work done by a force field in a moving particle around a closed path is zero.

A:

Q: Match the vector field with its graph. F(x, y) = yi -5+ -5+

A:

Q: Find the level surfaces of the scalar field U = ear, where the vectors are: a = (2, 2, 1) and r = =…

A:

Q: Is the vector field F = [yze"yz, xze#yz, xye"yz] conservative? %3D O Yes O No

A: The vector field is conservative if ∇×F→=0 therefore, find ∇×F→ ∇×F→=ijk∂∂x∂∂y∂∂zyzexyzxzexyzxyexyz

Q: Why does a conservative vector field produce zero circulationaround a closed curve?

A: Let F be an arbitrary conservative field vector around a closed curve C. To prove: Let A and B be…

Q: Match the vector field with its graph. F(x, y) = xi + 3yj 5 5 5 -5+ -5+

A:

Q: Find the curl and divergence for the given vector field.

A: →Divergence is a measure of spread of vector field at a point.→Curl describes the infinitesimal…

Q: What is the curl of a vector field? How can you interpret it?

A:

Q: Calculate the flux of the vector field F 6i + 2x2j- 2k, through the square of side 4 in the plane y…

A: Given vector field is F→=6i→+2x2j→−2k→ We have to find the flux of the given vector field. The flux…

Q: Knowing that the vector field given by the image function is conservative, determine its potential…

A:

Q: 1. Is the vector field V (x, y, z) = (2ry, x² – 2², –2yz) conservative?

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: What is cural of a vector field?how can you interpret it?

A: Let F=M i+N j+P k be a vector field. Where, M, N and P are continuously differentiable functions,

Q: Show that the vector field A (3x?z + y?)a,+ 2xya, + x³a, is conservative. %3D

A: Given problem:- Show that the vector field A = (3x²z+ y²)ax + 2xyay + x³az is conservative. =

Q: Match the planar vector field with the corresponding plot.

A: To draw a vector field of a vector  We can write the given vector as

Q: Select the option that gives the vector field that could correspond to this arrow map.

A: The vector field is when a vector is assigned to each point in the given space. The cartesian form…

Q: Match the planar vector field with the corresponding plot. (a) (4, x) y -6-

A:

Q: Match the vector field with its graph. F(x, y) = xi + 3yj 5- 5 -5+ -5+

A: Given data÷

Q: Sketch some vectors in the vector field F(x, y) = −yi + xj.

A:

Q: Sketch the vector field F(x, y) = 3i + 5j and calculate the flux then the circulation of F| along…

A: (1) The equation of line segment is: x+3-2+3=y-1-5-1=tx+3=t⇒x=t-3y-1=-6t⇒y=-6t+1 The curve…

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: