Verify that the vector field f is conservative and find its scalar potential. 3r f = + yz + 2xz) i+ + az)j+ (ry + a +3) k

Q: Express the vector field B = (x^2- y^2)ay + xzaz in spherical coordinates at ( 4, 30o, 120o) *…

A: Relation between cartesian and spherical co ordinate system x= r sinθ cosϕ   , y = r sinθ sinϕ   ,…

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

A:

Q: Compute the vector assigned to the points P = (1, 1) and Q = (-3, -3) by the vector field F = (-y,…

A:

Q: Find the directional derivative of the scalar field = x³y + 4xz at the point (1,-1, 2) along the…

A:

Q: Find the curl of the vector field F = curl F = 12 + j+ k

A:

Q: Compute the curl of the following vector field. 4r (4x, 4y, 4z) 7/2 (4x +4y +4z) " 2r The curl of F…

A:

Q: Sketch several representative vectors in the vector field given by F(x,y)-3yi-xj. b. a. 3 34

A: F(x,y)=3y i - x j F(0,1)=3(1) i - (0) j=3iF(1,0)=3(0) i - (1) j=-jF(-1,0)=3(0) i - (-1)…

Q: d) Use the curl test to determine if the following vector fields are conservative on R². If the test…

A: As per our guideline we can solve only first three subparts question. Let a vector field    F=…

Q: Find the curl of the vector fleld F(z, y) = 9ye" i+2ejat the point (-2, 0). Enter an exact answer.…

A: F(x, y)=9y2 ex3 i +2x2 ey jcurl F→=∇×F→curl F→=ijk∂∂x∂∂y∂∂z9y2 ex32x2 ey0         =∂∂x(2x2…

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: Find the directional derivative of the scalar field = xy + 4xz at the point (2, -2, 4) along the…

A:

Q: Find the flux of the vector field V(x, y, z) = 9xy²i + 9x²yj + z³k out of the unit sphere.

A:

Q: d) An open surface F is given by F = {(x,y, z) |z = 4 – x² – y² , z > 0} . Compute the flux of V…

A: V(x,y,z) = <2x+3y,2y+3x,-4z>F={ (x,y,z) : z=4-x2-y2 , z≥0}Consider, g(x,y,z) = z+x2+y2-4…

Q: (1 point) Show that the vector field F-yz Ч + zz 1 J-zuz 2 k is conservative by finding a scalar…

A: To find the scalar potential function f, assume

Q: Consider the vector field given by F(x, y) = (6xy – 152 y)i + (3a? + 3y – 10a°y)j find a function…

A:

Q: F = (x² - y)i + + (4z)j + (x²)k Find the curl (curl calculation) of the vector field?

A:

Q: (xy³ +3x²y, 3xy² + x²) 2+x ху у,

A: According to question given that Vector Field F⟨xy3+3x2y, 3xy2+x2⟩

Q: Find a potential function of the vector field. F(x,y) = (3x²y + 1, x³+2y)

A: we have to find potential function.

Q: Show that the vector field F = (yz² ye, rz² - e, 2xyz) is conservative, and find a potential…

A:

Q: Sketch the vector field: F(x, y) = (1, 2y) Carefully draw at least four vectors in each of the four…

A:

Q: A constant field vector is given by E = -5a_p + 10a_0 + 3a_z. Calculate the vector component of E at…

A: The problem is related to vector. The topic is multidimensional geometry. We find the component of…

Q: FTTT 10 5 10- -5- -10- X TTTTI 1C d.

A: We draw the sketch of vector field with the help of direction and length of vectors .

Q: Compute the curl of the vector field F = -4zi - 5y 3-4x k. =url= =

A: Given,  The Vector Field,F→=-4zi-5yj-4xk The Curl of the vector field is to be calculated.

Q: Use logarithmic differentiation to find the derivative of y

A: Using the properties of logarithmic functions, it can be written that: lnpq=ln p-ln qln pq=qln p…

Q: Determine if the scalar vector field is Lipschitz Continous on the indicated domain. f(t,x) = SiN…

A: The detailed solution is provided below.Explanation:By Mean value theorem,  we have to test whether…

Q: re f an -2, 3)

A:

Q: Let a vector field F(x,y)=sinx 1 + xy j be given. What is the integral representing the work done by…

A: Acoording to expert guidelines, I am allowed to answer first question only. Kindly repost the rest…

Q: ) Show that the vector field, F3 yz £ + zx ŷ + xy 2 c

A: Given: The vector field is, F3 = yz x^ + zx y^ + xy z^

Q: Show that the vector field A and B are everywhere perpendicular if A (2.-5,6) and B (6,5,-2)

A:

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: