Vector analysis

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Problem: Find the gradient of the scalar field

u (x, y, z) =

z e^{x^{2} + y^{2} + z^{2}}

at an arbitrary point and at the point

M_{0} (0, 0, 0)

. For the resulting vector field

\bar{a} =

Δ u (x, y, z)

div \bar{a}

rot \bar{a}

M_{0}

1.12.1 Vector analysis

3.81 $

Problem: Show that the gradient field of the scalar field

φ = 6 x^{3} y - 2 x y^{4} + z^{4} x^{2} y

is irrotational.

1.12.3 Vector analysis

0 $

Problem: Find the vector lines of the field

\vec{a} = (z - y) \vec{ı} +

+ (z + 1) \vec{ȷ} + (y + 1) \vec{k}

1.12.4 Vector analysis

1.27 $

Problem: Calculate the gradient of the scalar field

φ (x, y, z)

, find and draw level surfaces

φ = 0; \pm 1; \pm 3, where φ (x, y, z) = \frac{12 z}{x^{2} + y^{2}} .

1.12.5 Vector analysis

2.54 $

Problem: Given a vector field

\vec{a} = (2 x \vec{ı} + 3 y \vec{ȷ} + z \vec{k}) \cos | \vec{r} |

. Calculate scalar and vector products: 1.

(\nabla \cdot \vec{a}), 2

[\nabla \times \vec{a}]

1.12.6 Vector analysis

3.81 $

Problem: Calculate the gradient of the scalar field

φ (x, y, z)

, find and draw level surfaces

φ = 0; \pm 1; \pm 3, where φ (x, y, z) = \frac{3 y^{2} + x}{z^{2}} .

1.12.7 Vector analysis

2.54 $

Problem: Find the vector lines of gradient field of scalar function:

φ = y + y z - \frac{1}{2} x^{2} - z .

1.12.8 Vector analysis

2.54 $

Problem: For an arbitrary scalar field

u

div (grad u) = Δ u

1.12.9 Vector analysis

1.27 $

Problem: For any scalar field

u

div (u grad u)

rot (u grad u)

1.12.10 Vector analysis

3.05 $

Problem: Find the gradient of the scalar field

U (x, y, z) =

\arctan (\frac{z}{y + z})

at an arbitrary point and at the point

M_{0} = (1, 1, 0)

. For the resulting vector field

\bar{a} =

\nabla U (x, y, z)

div \bar{a}

rot \bar{a}

M_{0}

1.12.2 Vector analysis

3.81 $