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 \( \bar{a}= \) \( \Delta u(x, y, z) \) find \( \operatorname{div} \bar{a} \) and \( \operatorname{rot} \bar{a} \) at the point \( M_{0} \).

1.12.1 Vector analysis

3.57 $

Problem: Show that the gradient field of the scalar field \( \varphi=6 x^{3} y-2 x y^{4}+z^{4} x^{2} y \) is irrotational.

1.12.3 Vector analysis

0 $

Problem: Find the vector lines of the field \( \vec{a}=(z-y) \vec{\imath}+ \) \( +(z+1) \vec{\jmath}+(y+1) \vec{k} \).

1.12.4 Vector analysis

1.19 $

Problem: Calculate the gradient of the scalar field \( \varphi(x, y, z) \), find and draw level surfaces \[ \varphi=0 ; \pm 1 ; \pm 3, \quad \text { where } \quad \varphi(x, y, z)=\frac{12 z}{x^{2}+y^{2}} . \]

1.12.5 Vector analysis

2.38 $

Problem: Given a vector field \( \vec{a}=(2 x \vec{\imath}+3 y \vec{\jmath}+z \vec{k}) \cos |\vec{r}| \). Calculate scalar and vector products: 1. \( (\nabla \cdot \vec{a}), 2 \). \( [\nabla \times \vec{a}] \).

1.12.6 Vector analysis

3.57 $

Problem: Calculate the gradient of the scalar field \( \varphi(x, y, z) \), find and draw level surfaces \[ \varphi=0 ; \pm 1 ; \pm 3, \quad \text { where } \quad \varphi(x, y, z)=\frac{3 y^{2}+x}{z^{2}} . \]

1.12.7 Vector analysis

2.38 $

Problem: Find the vector lines of gradient field of scalar function: \[ \varphi=y+y z-\frac{1}{2} x^{2}-z . \]

1.12.8 Vector analysis

2.38 $

Problem: For an arbitrary scalar field \( u \) prove that \( \operatorname{div}(\operatorname{grad} u)=\Delta u \).

1.12.9 Vector analysis

1.19 $

Problem: For any scalar field \( u \) calculate: a) \( \operatorname{div}(u \operatorname{grad} u) \), b) \( \operatorname{rot}(u \operatorname{grad} u) \).

1.12.10 Vector analysis

2.85 $

Problem: Find the gradient of the scalar field \( U(x, y, z)= \) \( \arctan \left(\frac{z}{y+z}\right) \) at an arbitrary point and at the point \( M_{0}=(1,1,0) \). For the resulting vector field \( \bar{a}= \) \( \nabla U(x, y, z) \) find \( \operatorname{div} \bar{a} \) and \( \operatorname{rot} \bar{a} \) at the point \( M_{0} \).

1.12.2 Vector analysis

3.57 $