Show that the directions of the isoline and the gradient line at any given points in a scalar field are orthogonal to each other.

Q: d) Having seen how the inner product behaves under rotation we can now investigate its behaviour…

A: Given that a vectorvi​transforms under the Euclidean group as: vi​→Oij​vj​+ai​(4)where the…

Q: Answer 5,6,7,8,9 write whether attract or repel for every number

A: As we know that that  same pole repels and opposite poles attract i.e, N pole  will  repel N pole…

Q: Consider a rectangular surface of length L and width W in the xy plane with its center at the…

A:

Q: Find the k-component of (curl F) for the following vector field on the plane. F = (xe)i + (8y ex)j…

A: The vector field on the plane is given as F=xeyi^+8yexj^

Q: A vector field is defined by v = (yz, xz, xy). Show that curl v = 0.

A: v=(yz,xz,xy)

Q: Express the vector field W=(x^2-y^2)i +xzk in spherical coordinates at P(4,30degrees,120degrees).

A:

Q: Determine the origin ∇ · F and the vorticity ∇ × F of the vector field F(x, y, z) = (x + 2y − 3z)i +…

A:

Q: Use the law of sines and the law of cosines, in conjunction with sketches of the force polygons, to…

A:

Q: Considering the following points in rectangular coordinates: P; (0,0,2) and P2(0, v2, v2), determine…

A: The given function F is in spherical coordinates, and the line integral of the given function is…

Q: Find the arc length of the space curve given by the vector function r (t) t3 i + t2 j + (2t + 1) k…

A:

Q: Find a unit vector normal to the surface xy? + xz = 1 at the point (–1, 1, 1).

A: Given surface-xy2+xz=1P=(-,1,1)

Q: Q3- (a) Show that for what values of X,Y and Z the transformation of A = is orthogonal and then show…

A:

Q: y extend this idea to cylindrical o, z), which you may think of ar coordinate system with the cked…

A:

Q: Show that the vectors a = 2i+4j−8k, b = 3i−j+3k and c = −5i+11j−25k are coplanar.

A:

Q: Calculate the commutators [^px2, f(^x

A: Find commutator of px with x^2

Q: Given the vector field v = (xy²)t + Gy³z²)j – (x³yz)k determine if it is converging, diverging or…

A:

Q: Solve with explanation and calculation

A: By multiplying normal vector by -1 doesn't change anything for the plane.you can also reduce the…

Q: Given a scalar field with a function of f = Φ + Φz Calculate the slope in the direction of vector A…

A: Here we have a question from mathematical physics especially from vector calculus. As indicated by…

Q: The vector field G is described by the vector function shown below. Evaluate G at point P(-3, 4, 5)…

A: We have a given vector field G and we need to evaluate G at point P.  So, we need to put x = -3 ,…

Q: are orthogonal.

A: Given that, A=4-3-25 and B=150k The condition of the orthogonality of two vector is given as, A⨥B=0

Q: Get the expression for the divergent of a vector field in spherical coordinates.

A: Given : Divergence :                           Divergence of a vector field is a measure of the “…

Q: x²y°z* , e" sin(y) In(z). 2.3.4 g(x, y, z) = h(x, y, z)

A:

Q: Express the vector field D = (x² + y²)−¹(xɑx + yay) in cylindrical components and cylindrical…

A: The cylindrical coordinates (ρ, ϕ, z) of a vector are related to its cartesian components (x, y, z)…

Q: Given a vector field A = 5x ^ 2(sin (pi*x)/2) i, find V.A at x = 1

A: Vector A = 5x².(sin (pi.x)/2)i

Q: (b) Calculate Curl and divergence of given vector field x³y2 F = (3x + 2z?)î + ;- (z – 7x)k -j – (z…

A: F→=3x+2z2i^+x3y2zj^-z-7xk^ Curl:∇×F→∇×F→=i^j^k^∂∂x∂∂y∂∂z3x+2z2x3y2z-z+7x…

Q: 1 Consider the plane passing through the three points O, A, B, given by Oi + 0j + Ok A = i+2j + 3k…

A: First, we find two vectors that are in the plane using the coordinates of the points given. Then, we…

Q: Consider the vector field ʊ(r) = (x² + y²)êx + (x² + y²)êy + z²êz. Decompose the vector field (r)…

A: The vector field depicted in the image is described as F(r) = (x² + y²)êx + (x² + y²)êy + z²êz. The…

Q: (a) Let F₁ = x² 2 and F₂ = x Ŷ + y ŷ + z 2. Calculate the divergence and curl of F₁ and F₂. Which…

A:

Q: честоп 1 > frond. (y+z)i + (x+₂)j + (x+y) û -Ġ= 1²¹² i + y² G 1`72 - M = j yz ² lyzi + 2xz j + 3xy 2…

A: We have given some vectors then we have to find the divergence of these vectors. The required…