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In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given tensor field along a given curve. For example, the line integral over a scalar field (rank 0 tensor) can be interpreted as the area under the field carved out by a particular curve. This can be visualized as the surface created ...
The classical Stokes' theorem relates the surface integral of the curl of a vector field over a surface in Euclidean three-space to the line integral of the vector field over its boundary. It is a special case of the general Stokes theorem (with n = 2 {\displaystyle n=2} ) once we identify a vector field with a 1-form using the metric on ...
e. Pakistan studies curriculum (Urdu: مطالعہ پاکستان Muṭāla-e-Pākistān) is the name [1][2] of a curriculum of academic research and study that encompasses the culture, demographics, geography, history, International Relations and politics of Pakistan. The subject is widely researched in and outside the country, though outside ...
Calculus. In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate over this surface a scalar field (that is, a function of position which returns a ...
H-8/4, Islamabad. Website. fbise.edu.pk. The Federal Board of Intermediate and Secondary Education (Urdu: وفاقی ہیئت برائے متوسط و ثانوی تعلیم), commonly known as FBISE, is an autonomous board under the Ministry of Federal Education for examining the intermediate and secondary education in Pakistan and abroad for ...
The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a conservative force. By placing φ as potential, ∇φ is a conservative field. Work done by conservative forces does not depend on the path followed by the object, but only the end points, as ...
The classical theorem of Stokes can be stated in one sentence: The line integral of a vector field over a loop is equal to the surface integral of its curl over the enclosed surface. Stokes' theorem is a special case of the generalized Stokes theorem. [5][6] In particular, a vector field on can be considered as a 1-form in which case its curl ...
Education in Pakistan is overseen by the Federal Ministry of Education and the provincial governments, while the federal government mostly assists in curriculum development, accreditation and the financing of research and development. Article 25-A of the Constitution of Pakistan obligates the state to provide free and compulsory quality ...